Prime groupoid graded rings with applications to partial skew groupoid rings

Abstract

In this paper, we investigate primeness of groupoid graded rings. We provide a set of necessary and sufficient conditions for primeness of a nearly-epsilon strongly groupoid graded ring. Furthermore, we apply our main result to get a characterization of prime partial skew groupoid rings, and in particular of prime groupoid rings, thereby generalizing a classical result by Connell and partially generalizing recent results by Steinberg.

KEYWORDS:

• Group-type partial action
• groupoid graded ring
• nearly epsilon-strongly groupoid graded ring
• partial skew groupoid ring
• prime ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W50
• 16N60
• 16S35
• 16S99
• 16W10

1 Introduction

Throughout this paper, all rings are assumed to be associative but not necessarily unital. Recall that a ring S is said to be prime if there are no nonzero ideals I, J of S such that $\mathit{\text{IJ}}=\left\{0\right\}.$

In 1963, Connell [7, Theorem 8] gave a characterization of prime group rings. Indeed, given a unital ring A and a group G, the corresponding group ring $A[G]$ is prime if, and only if, A is prime and G has no nontrivial finite normal subgroup.

Given a group G, recall that a ring S is said to be G-graded if there is a collection ${\left\{S_g\right\}}_{g\in G}$ of additive subgroups of S such that $S={\oplus }_{g\in G}{S}_g$, and $S_g{S}_h\subseteq S_{\mathit{\text{gh}}}$, for all $g,h\in G$. If, in addition, $S_g{S}_h=S_{\mathit{\text{gh}}}$, for all $g,h\in G$, then S is said to be strongly G-graded. The class of unital strongly G-graded rings includes for instance all group rings, all twisted group rings, and all G-crossed products.

In 1984, Passman [22, Theorem 1.3] generalized Connell’s result by giving a characterization of prime unital strongly group graded rings. In recent years, various generalizations of strongly group graded rings have appeared in the literature. In [16], so-called nearly epsilon-strongly group graded rings were introduced. That class of rings contains for instance all unital strongly group graded rings, all epsilon-strongly group graded rings, all Leavitt path algebras, and all unital partial crossed products (see [16, 17]). In [9], a characterization of prime nearly epsilon-strongly group graded rings was established, thereby generalizing Passman’s result to a non-unital and non-strong setting.

In this paper, we turn our focus to rings graded by groupoids. Let G be a groupoid. Recall that a ring S is said to be G-graded if there is a collection ${\left\{S_g\right\}}_{g\in G}$ of additive subgroups of S such that $S={\oplus }_{g\in G}{S}_g$, and $S_g{S}_h\subseteq S_{\mathit{\text{gh}}}$ whenever $g,h\in G$ are composable, and $S_g{S}_h=\left\{0\right\}$ otherwise. Groupoid rings, groupoid crossed products, and partial skew groupoid rings are examples of rings that, by construction, are naturally graded by groupoids (see e.g. [1, 2, 4, 5, 20]). Partial skew groupoid rings play a key role in the theory of partial Galois extensions for partial groupoid actions (see e.g., [3, Theorem 5.3]) and, in particular, in the Galois theory of weak Hopf algebra actions on algebras (see [6]). Some crossed product algebras defined by separable extensions are not, in a natural way, graded by groups, but instead by groupoids (see e.g. [11, 12]). Another concrete example of when it can be beneficial to make use of a groupoid grading instead of just a group grading is in the study of Leavitt path algebras. There, one may utilize the canonical grading by the so-called free path groupoid which is finer, and encodes more of the structure, than the coarser canonical $\mathbb{Z}$-grading (see [8]).

Suppose that G is a groupoid and that S is a G-graded ring. Following [10], we shall say that S is nearly epsilon-strongly G-graded if, for each $g\in G, S_g{S}_{g^{-1}}$ is an s-unital ring and $S_g{S}_{g^{-1}}{S}_g=S_g$. Our main result provides a characterization of prime nearly epsilon-strongly groupoid graded rings.

Theorem 1.1.

Let G be a groupoid, let S be a nearly epsilon-strongly G-graded ring, and let $G\prime :=\{g\in G:S_{s(g)}\ne \{0\}\text{and}{S}_{r(g)}\ne \{0\left\}\right\}$. The following statements are equivalent:

1. S is prime;

2. ${\oplus }_{e\in G_0}{S}_e$ is G-prime, and for every $e\in G_0^{\prime },$ ${\oplus }_{g\in G_e^e}{S}_g$ is prime;

3. ${\oplus }_{e\in G_0}{S}_e$ is G-prime, and for some $e\in G_0^{\prime },$ ${\oplus }_{g\in G_e^e}{S}_g$ is prime;

4. S is graded prime, and for every $e\in G_0^{\prime },$ ${\oplus }_{g\in G_e^e}{S}_g$ is prime;

5. S is graded prime, and for some $e\in G_0^{\prime },$ ${\oplus }_{g\in G_e^e}{S}_g$ is prime;

6. For every $e\in G_0^{\prime },$ e is a support-hub, and ${\oplus }_{g\in G_e^e}{S}_g$ is prime;

7. For some $e\in G_0^{\prime },$ e is a support-hub, and ${\oplus }_{g\in G_e^e}{S}_g$ is prime.

Here $G_e^e$ denotes the isotropy group of an element $e\in G_0^{\prime }.$ For more details about the statements in the above theorem, see e.g. Definitions 3.17 and 3.21.

We point out that our main result reduces the primeness investigation for a groupoid graded ring to the group case. Indeed, ${\oplus }_{g\in G_e^e}{S}_g$ is a nearly epsilon-strongly group graded ring (cf. [16]). Hence, the main result of [9] can be used to decide whether it is prime.

Here is an outline of this paper. In Section 2, based on [15, 19, 25], we recall some basic definitions and properties about groupoids, groupoid graded rings, and s-unital rings that will be used throughout the paper. In Section 3, we record some basic properties of nearly epsilon-strongly groupoid graded rings. Inspired by [9], for such a ring S, we establish a relationship between the G-invariant ideals of ${\oplus }_{e\in G_0}{S}_e$ and the G-graded ideals of S (see Theorem 3.16). Moreover, we provide necessary conditions for graded primeness of S (see Section 3.3) and establish our main result which is a characterization of prime nearly epsilon-strongly groupoid graded rings (see Theorem 1.1). Finally, in Section 4, we apply our results to partial skew groupoid rings, skew groupoid rings, and groupoid rings. In particular, we give a characterization of prime partial skew groupoid rings associated with groupoid partial actions of group-type [4, 5] (see Theorem 4.15). Using that every global action of a connected groupoid is of group-type, we get a characterization of prime skew groupoid rings of connected groupoids (see Theorem 4.19). Furthermore, we establish a generalization of Connell’s classical result, by providing a characterization of prime groupoid rings (see Theorem 4.26).

2 Preliminaries

In this section, we recall some notions and basic notation regarding groupoids and graded rings.

2.1 Groupoids

By a groupoid, we shall mean a small category G in which every morphism is invertible. Each object of G will be identified with its corresponding identity morphism, allowing us to view G₀, the set of objects of G, as a subset of the set of morphisms of G. The set of morphisms of G will simply be denoted by G. This means that $G_0:=\{g{g}^{-1}:g\in G\}\subseteq G$.

The range and source maps $r,s:G\to G_0$, indicate the range (codomain) respectively source (domain) of each morphism of G. By abuse of notation, the set of composable pairs of G is denoted by $G^2:=\{(g,h)\in G\times G:s(g)=r(h)\}$. For each $e\in G_0,$ we denote the corresponding isotropy group by $G_e^e:=\{g\in G:s(g)=r(g)=e\}.$

Definition 2.1.

Let G be a groupoid.

1. G is said to be connected, if for every pair $e,f\in G_0$, there exists $g\in G$ such that s(g) = e and r(g) = f.

2. A nonempty subset H of G is said to be a subgroupoid of G, if $H^{-1}\subseteq H$ and $\mathit{\text{gh}}\in H$ whenever $g,h\in H$ and $(g,h)\in G^2$.

2.2 Groupoid graded rings

Definition 2.2.

Let G be a groupoid. A ring S is said to be G-graded (or graded by G) if there is a collection ${\left\{S_g\right\}}_{g\in G}$ of additive subgroups of S such that $S={\oplus }_{g\in G}{S}_g$, and $S_g{S}_h\subseteq S_{\mathit{\text{gh}}}$, if $(g,h)\in G^2$, and $S_g{S}_h=\left\{0\right\}$, otherwise.

Remark 2.3.

Suppose that G is a groupoid and that S is a G-graded ring.

1. If H is a subgroupoid of G, then $S_H:={\oplus }_{h\in H}{S}_h$ is an H-graded subring of S. In particular, note that ${\oplus }_{e\in G_0}{S}_e$ and ${\oplus }_{g\in G_e^e}{S}_g$ are subrings of S for every $e\in G_0$.

2. For any element $c=\sum _{g\in G}{c}_g\in S$, with $c_g\in S_g$, we define $\text{Supp}(c):=\{g\in G:c_g\ne 0\}$.

3. An ideal I of S is said to be a graded ideal (or G-graded ideal) if $I={\oplus }_{g\in G}(I\cap S_g)$.

The next lemma generalizes [19, Lemma 2.4]. For the convenience of the reader, we include a proof.

Lemma 2.4.

Let G be a groupoid and let S be a G-graded ring. Suppose that H is a subgroupoid of G. Define ${\pi }_H:S\to S_H$ by ${\pi }_H(\sum _{g\in G}{c}_g):=\sum _{h\in H}{c}_h.$ The following assertions hold:

1. The map ${\pi }_H:S\to S_H$ is additive.

2. If $a\in S$ and $b\in S_H$, then ${\pi }_H(\mathit{\text{ab}})={\pi }_H(a)b$ and ${\pi }_H(\mathit{\text{ba}})=b{\pi }_H(a)$.

Proof.

(i) This is clear.

(ii) Take $a\in S$ and $b\in S_H$. Put $a\prime :=a-{\pi }_H(a)$. Clearly, $a=a\prime +{\pi }_H(a)$ and $\text{Supp}(a\prime )\subseteq G\setminus H$. If $g\in G\setminus H$ and $h\in H$, then either the composition gh does not exist or it belongs to $G\setminus H$. Thus, $\text{Supp}(a\prime b)\subseteq \varnothing \cup G\setminus H$. Hence, ${\pi }_H(\mathit{\text{ab}})={\pi }_H((a\prime +{\pi }_H(a))b)={\pi }_H(a\prime b)+{\pi }_H({\pi }_H(a)b)=0+{\pi }_H(a)b.$ Analogously, one may show that ${\pi }_H(\mathit{\text{ba}})=b{\pi }_H(a)$. □

2.3 s-unital rings

We briefly recall the definitions of s-unital modules and rings as well as some key properties.

Definition 2.5

([15, cf. Definition 4]). Let R be a ring and let M be a left (resp. right) R-module. We say that M is s-unital if $m\in \mathit{\text{Rm}}$ (resp. $m\in \mathit{\text{mR}}$) for every $m\in M$. If M is an R-bimodule, then we say that M is s-unital if it is s-unital both as a left R-module and as a right R-module. The ring R is said to be left s-unital (resp. right s-unital) if it is left (resp. right) s-unital as a left (resp. right) module over itself. The ring R is said to be s-unital if it is s-unital as a bimodule over itself.

The following results are due to Tominaga [25]. For the proofs, we refer the reader to [15, Propositions 2.8 and 2.10].

Proposition 2.6.

Let R be a ring and let M be a left (resp. right) R-module. Then M is left (resp. right) s-unital if, and only if, for all $n\in {\mathbb{Z}}^{+}$ and all $m_1,\dots ,m_n\in M$ there is some $a\in R$ such that $a{m}_i=m_i$ (resp. $m_{i}a=m_i$) for every $i\in \{1,\dots ,n\}$.

Proposition 2.7.

Let R be a ring and let M be an R-bimodule. Then M is s-unital if, and only if, for all $n\in {\mathbb{Z}}^{+}$ and all $m_1,\dots ,m_n\in M$ there is some $a\in R$ such that $a{m}_i=m_{i}a=m_i$ for every $i\in \{1,\dots ,n\}$.

Remark 2.8.

The element a, in Proposition 2.7, is commonly referred to as an s-unit for the set $\{m_1,\dots ,m_n\}$.

3 Groupoid graded rings

Throughout this section, unless stated otherwise, G denotes an arbitrary groupoid.

3.1 Nearly epsilon-strongly groupoid graded rings

In this section, we will recall the notion of a nearly epsilon-strongly groupoid graded ring and record some of its basic properties.

Definition 3.1

([10, Definition 3]). Let S be a G-graded ring. We say that S is nearly epsilon-strongly G-graded if, for each $g\in G, S_g{S}_{g^{-1}}$ is an s-unital ring and $S_g{S}_{g^{-1}}{S}_g=S_g$.

Remark 3.2.

The above definition simultaneously generalizes [16, Definition 3.3] and [18, Definition 34].

The following characterization of a nearly epsilon-strongly groupoid graded ring appeared in [10, Proposition 15] without a proof. For the convenience of the reader, we provide it here.

Proposition 3.3.

Let S be a G-graded ring. The following statements are equivalent:

1. S is nearly epsilon-strongly G-graded;

2. For all $g\in G$ and $d\in S_g,$ there exist ${ϵ}_g(d)\in S_g{S}_{g^{-1}}$ and $ϵ{\prime }_g(d)\in S_{g^{-1}}{S}_g$ such that ${ϵ}_g(d)d=$ $dϵ{\prime }_g(d)=d.$

Proof.

Suppose that (i) holds. Let $g\in G$ and $d\in S_g.$ We may write $d={\sum }_{i=1}^n{a}_i{b}_i{c}_i$ for some $n\in {\mathbb{Z}}^{+}$, $a_1,\dots ,a_n,c_1,\dots ,c_n\in S_g$ and $b_1,\dots ,b_n\in S_{g^{-1}}$. Notice that $a_i{b}_i\in S_g{S}_{g^{-1}}$ and $b_i{c}_i\in S_{g^{-1}}{S}_g$ for every $i\in \{1,\dots ,n\}.$ By assumption, $S_g{S}_{g^{-1}}$ and $S_{g^{-1}}{S}_g$ are s-unital and hence, by Proposition 2.6, there exist ${ϵ}_g(d)\in S_g{S}_{g^{-1}}$ and $ϵ{\prime }_g(d)\in S_{g^{-1}}{S}_g$ such that ${ϵ}_g(d)a_i{b}_i=a_i{b}_i$ and $b_i{c}_iϵ{\prime }_g(d)=b_i{c}_i$ for every $i\in \{1,\dots ,n\}.$ Thus, ${ϵ}_g(d)d=dϵ{\prime }_g(d)=d.$ This shows that (ii) holds.

Conversely, suppose that (ii) holds. Let $g\in G.$ Note that, by assumption, S_g is s-unital as a left $S_g{S}_{g^{-1}}$-module and $S_{g^{-1}}$ is s-unital as a right $S_g{S}_{g^{-1}}$-module. Let $m\in S_g{S}_{g^{-1}}.$ We may write $m={\sum }_{i=1}^n{a}_i{b}_i$ for some $n\in {\mathbb{Z}}^{+},$ $a_1,\dots ,a_n\in S_g$ and $b_1,\dots ,b_n\in S_{g^{-1}}$. By s-unitality of the left $S_g{S}_{g^{-1}}$-module S_g, and Proposition 2.6, there is some $u\in S_g{S}_{g^{-1}}$ such that $u{a}_i=a_i$ for every $i\in \{1,\dots ,n\}.$ Similarly, there is some $u\prime \in S_g{S}_{g^{-1}}$ such that $b_{i}u\prime =b_i$ for every $i\in \{1,\dots ,n\}.$ Hence, $\mathit{\text{um}}=m$ and $\mathit{\text{mu}}\prime =m$. This shows that $S_g{S}_{g^{-1}}$ is s-unital. Note that $S_g{S}_{g^{-1}}{S}_g\subseteq S_{g{g}^{-1}g}=S_g.$ Using that S_g is s-unital as a left $S_g{S}_{g^{-1}}$-module we get that $S_g\subseteq (S_g{S}_{g^{-1}})S_g$. Thus, $S_g=S_g{S}_{g^{-1}}{S}_g$. This shows that (i) holds. □

Corollary 3.4.

Let S be a nearly epsilon-strongly G-graded ring and let a be a nonzero element of S. If $g\in \text{Supp}(a)$, then there are elements $b,c\in S_{g^{-1}}$ such that $a_{g}b\ne 0$ and $c{a}_g\ne 0$.

The following result generalizes [9, Proposition 2.13] from the group setting.

Proposition 3.5.

Let S be a nearly epsilon-strongly G-graded ring. The following assertions hold:

1. S_e is an s-unital ring, for every $e\in G_0$.

2. $d\in d({\oplus }_{e\in G_0}{S}_e)\cap ({\oplus }_{e\in G_0}{S}_e)d$, for every $d\in S.$

3. S is s-unital and ${\oplus }_{e\in G_0}{S}_e$ is an s-unital subring of S.

4. Suppose that H is a subgroupoid of G. Then ${\oplus }_{h\in H}{S}_h$ is a nearly epsilon-strongly H-graded ring.

5. ${\oplus }_{g\in G_e^e}{S}_g$ is an s-unital ring, for every $e\in G_0.$

6. The set

   $G\prime :=\{g\in G:S_{s(g)}\ne \{0\}\text{and}{S}_{r(g)}\ne \{0\left\}\right\}$

   is a subgroupoid of G.

7. $S={\oplus }_{g\in G\prime }{S}_g.$

Proof.

1. Take $e\in G_0$. By assumption, $S_e=S_e{S}_e{S}_e.$ Therefore, $S_e=(S_e{S}_e)S_e\subseteq S_e{S}_e\subseteq S_e$ and hence $S_e=S_e{S}_e$.

2. Let $d=\sum _{t\in G}{d}_t\in S$, with $d_t\in S_t.$ Take $g\in \text{Supp}(d)$. By Proposition 3.3, there exist ${ϵ}_g(d_g)\in S_g{S}_{g^{-1}}\subseteq S_{r(g)}$ and ${ϵ}_g^{\prime }(d_g)\in S_{g^{-1}}{S}_g\subseteq S_{s(g)}$ such that ${ϵ}_g(d_g)d_g=d_g{ϵ}_g^{\prime }(d_g)=d_g.$ The set $B:=\{r(t):t\in \text{Supp}(d)\}$ is finite, because $\text{Supp}(d)$ is finite. For every $f\in B$, by (i), S_f is s-unital, and we let $v_f\in S_f$ be an s-unit for the finite set $\{{ϵ}_t(d_t):t\in \text{Supp}(d)\text{and}r(t)=f\}\subseteq S_f.$ Define $a:=\sum _{f\in B}{v}_f\in {\oplus }_{e\in G_0}{S}_e.$ We get that

   $\mathit{\text{ad}}=\sum _{f\in B}{v}_f\sum _{t\in G}{d}_t=\sum _{t\in \text{Supp}(d)}{v}_{r(t)}{d}_t=\sum _{t\in \text{Supp}(d)}{v}_{r(t)}({ϵ}_t(d_t)d_t)$

   $=\sum _{t\in \text{Supp}(d)}(v_{r(t)}{ϵ}_t(d_t))d_t=\sum _{t\in \text{Supp}(d)}{ϵ}_t(d_t)d_t=\sum _{t\in G}{d}_t=d.$

   Similarly, we define the finite set $D:=\{s(t):t\in \text{Supp}(d)\}.$ For every $f\in D,$ by (i), we let $v_f^{\prime }\in S_f$ be an s-unit for the finite set $\{{ϵ}_t^{\prime }(d_t):t\in \text{Supp}(d)\text{and}s(t)=f\}\subseteq S_f.$ Define $a\prime :=\sum _{f\in D}{v}_f\prime \in {\oplus }_{e\in G_0}{S}_e.$ We get that

   $\mathit{\text{da}}\prime =\sum _{t\in G}{d}_t\sum _{f\in D}{v}_f\prime =\sum _{t\in \text{Supp}(d)}{d}_t{v}_{s(t)}^{\prime }=\sum _{t\in \text{Supp}(d)}(d_t{ϵ}_t^{\prime }(d_t))v_{s(t)}^{\prime }$

   $=\sum _{t\in \text{Supp}(d)}{d}_t({ϵ}_t^{\prime }(d_t)v_{s(t)}^{\prime })=\sum _{t\in \text{Supp}(d)}{d}_t{ϵ}_t^{\prime }(d_t)=\sum _{t\in G}{d}_t=d.$

3. It follows immediately from (ii).

4. It follows immediately from Remark 2.3 and the fact that $H\subseteq G$.

5. Take $e\in G_0$. Clearly, the isotropy group $G_e^e$ is a subgroupoid of G. By (iv) and (iii) we get that ${\oplus }_{g\in G_e^e}{S}_g$ is s-unital.

6. Clearly, $g^{-1}\in G\prime$ whenever $g\in G\prime$. Suppose that $g,h\in G\prime$ and $(g,h)\in G^2$. Then $S_{s(\mathit{\text{gh}})}=S_{s(h)}\ne \left\{0\right\}$ and $S_{r(\mathit{\text{gh}})}=S_{r(g)}\ne \left\{0\right\}.$ Therefore, $\mathit{\text{gh}}\in G\prime .$ This shows that $G\prime$ is a subgroupoid of G.

7. Take $g\in G$ such that $S_g\ne \left\{0\right\}$. We claim that $g\in G\prime$. If we assume that the claim holds, then clearly $S={\oplus }_{g\in G\prime }{S}_g.$ Now we show the claim. Let $d\in S_g$ be nonzero. By Proposition 3.3, there are ${ϵ}_g(d)\in S_g{S}_{g^{-1}}\subseteq S_{r(g)}$ and ${ϵ}_g^{\prime }(d)\in S_{g^{-1}}{S}_g\subseteq S_{s(g)}$ such that ${ϵ}_g(d)d=d{ϵ}_g^{\prime }(d)=d.$ In particular, $S_{r(g)}\ne \left\{0\right\}$ and $S_{s(g)}\ne \left\{0\right\}.$ □

Remark 3.6.

(a) Note that (vi) above holds for any G-graded ring. For (vii), however, the nearly epsilon-strongness of the G-grading is used.

(b) Suppose that S is a nearly epsilon-strongly G-graded ring. By Proposition 3.5(vii), $g\in G\prime$ whenever $S_g\ne \left\{0\right\}$. The converse, however, need not hold (see e.g., Example 4.22).

Example 3.7.

Let $G:=\{f_1,f_2,f_3,g,h,g^{-1},h^{-1},h{g}^{-1},g{h}^{-1}\}$ be a groupoid with $G_0=\{f_1,f_2,f_3\}$ and depicted as follows:

Let $S:={\mathcal{M}}_3(\mathbb{Z}),$ be the ring of 3 × 3 matrices over $\mathbb{Z},$ and let ${\left\{e_{\mathit{\text{ij}}}\right\}}_{i,j}$ denote the standard matrix units. We define: $$\begin{array}{cccc}{S}_g\hfill & :=\mathbb{Z}{e}_{12},\hfill & S_{g^{-1}}:=\mathbb{Z}{e}_{21},\hfill & S_h:=\mathbb{Z}{e}_{32},\hfill \\ S_{h^{-1}}\hfill & :=\mathbb{Z}{e}_{23},\hfill & S_{g{h}^{-1}}:=\mathbb{Z}{e}_{13},\hfill & S_{h{g}^{-1}}:=\mathbb{Z}{e}_{31},\hfill \\ S_{f_1}\hfill & :=\mathbb{Z}{e}_{11},\hfill & S_{f_2}:=\mathbb{Z}{e}_{22},\hfill & S_{f_3}:=\mathbb{Z}{e}_{33}.\hfill \end{array}$$

Notice that $S={\oplus }_{l\in G}{S}_l.$ It is not difficult to see that the ring ${\mathcal{M}}_3(\mathbb{Z})$ is nearly epsilon-strongly G-graded.

3.2 Invariance in groupoid graded rings

Inspired by [9, Sections 3–4], we shall now examine the relationship between G-graded ideals of a G-graded ring S and G-invariant ideals of the subring ${\oplus }_{e\in G_0}{S}_e.$ Throughout this section, S denotes an arbitrary G-graded ring.

Definition 3.8

([9, Definitions 3.1 and 3.3]). Let S be a G-graded ring.

1. For any $g\in G$ and any subset I of S, we write $I^g:=S_{g^{-1}}I{S}_g.$

2. Let H be a subgroupoid of G and let I be a subset of S. Then, I is called H-invariant if $I^g\subseteq I$ for every $g\in H.$

Remark 3.9.

Note that if $g\in G,$ and $I\subseteq S,$ then $$I^g=\left\{\sum _{k=1}^n{a}_k{x}_k{b}_k:n\in {\mathbb{Z}}^{+},a_k\in S_{g^{-1}},x_k\in I\text{and}{b}_k\in S_g\text{for each}k\in \{1,\dots ,n\}\right\}.$$

Lemma 3.10.

If $g\in G$ and J is an ideal of ${\oplus }_{e\in G_0}{S}_e,$ then J^g is an ideal of ${\oplus }_{e\in G_0}{S}_e.$

Proof.

Let $g\in G$ and let J be an ideal of ${\oplus }_{e\in G_0}{S}_e$. Notice that J^g is an additive subgroup of ${\oplus }_{e\in G_0}{S}_e.$ Moreover, $({\oplus }_{e\in G_0}{S}_e)J^g=({\oplus }_{e\in G_0}{S}_e)(S_{g^{-1}}J{S}_g)=S_{s(g)}{S}_{g^{-1}}J{S}_g\subseteq S_{g^{-1}}J{S}_g=J^g.$ Analogously, $J^g({\oplus }_{e\in G_0}{S}_e)\subseteq J^g.$ □

Proposition 3.11.

Suppose that J is an ideal of ${\oplus }_{e\in G_0}{S}_e.$ Then SJS is a G-graded ideal of S.

Proof.

It is clear that SJS is an ideal of S and that ${\oplus }_{g\in G}((\mathit{\text{SJS}})\cap S_g)\subseteq \mathit{\text{SJS}}.$ Now, we show the reversed inclusion. Take $g,h\in G, a_g\in S_g,$ $c_h\in S_h$ and $b=\sum _{e\in G_0}{b}_e\in J.$ If $s(g)\ne r(h),$ then $a_{g}b{c}_h=0\in (\mathit{\text{SJS}})\cap S_{\mathit{\text{gh}}}.$ Otherwise, $s(g)=r(h)$, and then $a_{g}b{c}_h=a_g{b}_{s(g)}{c}_h\in (\mathit{\text{SJS}})\cap S_{\mathit{\text{gh}}}.$ Thus, $\mathit{\text{SJS}}={\oplus }_{g\in G}((\mathit{\text{SJS}})\cap S_g).$ □

Lemma 3.12.

Suppose that ${\oplus }_{e\in G_0}{S}_e$ is s-unital and that J is an ideal of ${\oplus }_{e\in G_0}{S}_e.$ Then J is G-invariant if, and only if, $(\mathit{\text{SJS}})\cap {\oplus }_{e\in G_0}{S}_e=J.$

Proof.

We first show the “only if” statement. Suppose that J is G-invariant. For each $e\in G_0,$ we have $$(\mathit{\text{SJS}})\cap S_e\subseteq \left(\sum _{\begin{array}{c}g\in G\\ s(g)=e\end{array}}(S_{g^{-1}}J{S}_g)\right)\cap S_e\subseteq \left(\sum _{\begin{array}{c}g\in G\\ s(g)=e\end{array}}{J}^g\right)\cap S_e\subseteq J.$$

Let $a=\sum _{f\in G_0}{a}_f\in (\mathit{\text{SJS}})\cap {\oplus }_{e\in G_0}{S}_e$. By Proposition 3.11, SJS is G-graded and we notice that $a_f\in (\mathit{\text{SJS}})\cap S_f\subseteq J$. Thus, $(\mathit{\text{SJS}})\cap {\oplus }_{e\in G_0}{S}_e\subseteq J$. By assumption, ${\oplus }_{e\in G_0}{S}_e$ is s-unital and $J\subseteq {\oplus }_{e\in G_0}{S}_e$. Hence, $J\subseteq ({\oplus }_{e\in G_0}{S}_e)J({\oplus }_{e\in G_0}{S}_e)\subseteq (\mathit{\text{SJS}})\cap {\oplus }_{e\in G_0}{S}_e.$

Now we show the “if” statement. Suppose that $(\mathit{\text{SJS}})\cap {\oplus }_{e\in G_0}{S}_e=J.$ Take $g\in G$ and notice that $J^g=S_{g^{-1}}J{S}_g\subseteq (\mathit{\text{SJS}})\cap S_{s(g)}\subseteq (\mathit{\text{SJS}})\cap {\oplus }_{e\in G_0}{S}_e=J.$ Thus, J is G-invariant. □

Lemma 3.13.

If I is a G-graded ideal of S, then $I\cap {\oplus }_{e\in G_0}{S}_e$ is a G-invariant ideal of ${\oplus }_{e\in G_0}{S}_e.$

Proof.

Let I be a G-graded ideal of S. Clearly, $I\cap {\oplus }_{e\in G_0}{S}_e$ is an ideal of ${\oplus }_{e\in G_0}{S}_e.$ Take $g\in G.$ Notice that $S_{g^{-1}}(I\cap S_{r(g)})S_g\subseteq I\cap S_{s(g)}.$ Furthermore, if $e\in G_0\setminus \{r(g)\},$ then $S_{g^{-1}}(I\cap S_e)S_g=\left\{0\right\}.$ Therefore, ${(I\cap {\oplus }_{e\in G_0}{S}_e)}^g=S_{g^{-1}}(I\cap {\oplus }_{e\in G_0}{S}_e)S_g\subseteq I\cap {\oplus }_{e\in G_0}{S}_e.$ □

Lemma 3.14.

Let S be a nearly epsilon-strongly G-graded ring. If I is a G-graded ideal of S, then $I=(I\cap {\oplus }_{e\in G_0}{S}_e)S=S(I\cap {\oplus }_{e\in G_0}{S}_e)=S(I\cap {\oplus }_{e\in G_0}{S}_e)S.$

Proof.

Let I be a G-graded ideal of S. By Proposition 3.5, S is s-unital and hence $(I\cap {\oplus }_{e\in G_0}{S}_e)\subseteq (I\cap {\oplus }_{e\in G_0}{S}_e)S.$ Thus, $S(I\cap {\oplus }_{e\in G_0}{S}_e)\subseteq S(I\cap {\oplus }_{e\in G_0}{S}_e)S\subseteq I.$ Analogously, $(I\cap {\oplus }_{e\in G_0}{S}_e)S\subseteq S(I\cap {\oplus }_{e\in G_0}{S}_e)S\subseteq I.$

We claim that $I\subseteq S(I\cap {\oplus }_{e\in G_0}{S}_e).$ Take $g\in G$ and $a_g\in I\cap S_g.$ By Proposition 3.3, there is some ${ϵ}_g(a_g)\in S_g{S}_{g-1}$ such that $a_g={ϵ}_g(a_g)a_g.$ Then, ${ϵ}_g(a_g)={\sum }_{j=1}^n{b}_j{c}_j$ for some $n\in {\mathbb{Z}}^{+}, b_1,\dots ,b_n\in S_g$ and $c_1,\dots ,c_n\in S_{g^{-1}}.$ Notice that $c_j{a}_g\in (S_{g^{-1}}{S}_g)\cap I\subseteq S_{s(g)}\cap I$ for every $j\in \{1,\dots ,n\}.$ Hence, $a_g={ϵ}_g(a_g)a_g={\sum }_{j=1}^n{b}_j{c}_j{a}_g\in S_g(S_{s(g)}\cap I)\subseteq S({\oplus }_{e\in G_0}{S}_e\cap I).$ Using that I is G-graded, we get that $I\subseteq S({\oplus }_{e\in G_0}{S}_e\cap I).$ Similarly, $I\subseteq ({\oplus }_{e\in G_0}{S}_e\cap I)S.$ Thus, $I=S({\oplus }_{e\in G_0}{S}_e\cap I)=({\oplus }_{e\in G_0}{S}_e\cap I)S$ and $I=S({\oplus }_{e\in G_0}{S}_e\cap I)\subseteq S({\oplus }_{e\in G_0}{S}_e\cap I)S\subseteq I.$ □

Corollary 3.15.

Let S be a nearly epsilon-strongly G-graded ring. If J is a G-invariant ideal of ${\oplus }_{e\in G_0}{S}_e,$ then $\mathit{\text{SJS}}=\mathit{\text{JS}}=\mathit{\text{SJ}}.$

Proof.

Let J be a G-invariant ideal of ${\oplus }_{e\in G_0}{S}_e.$ By Proposition 3.11, SJS is a G-graded ideal of S, and, by Lemma 3.14, $\mathit{\text{SJS}}=((\mathit{\text{SJS}})\cap {\oplus }_{e\in G_0}{S}_e)S=S((\mathit{\text{SJS}})\cap {\oplus }_{e\in G_0}{S}_e).$ Thus, by Proposition 3.5(iii) and Lemma 3.12, $\mathit{\text{SJS}}=\mathit{\text{JS}}=\mathit{\text{SJ}}.$ □

By Lemmas 3.13 and 3.11, the following maps are well defined: $$\varphi :\left\{G\text{‐graded ideals of}S\right\}\ni I\mapsto I\cap {\oplus }_{e\in G_0}{S}_e\in \left\{G\text{‐invariant ideals of}{\oplus }_{e\in G_0}{S}_e\right\}$$ $$\psi :\left\{G\text{‐invariant ideals of}{\oplus }_{e\in G_0}{S}_e\right\}\ni J\mapsto \mathit{\text{SJS}}\in \left\{G\text{‐graded ideals of}S\right\}.$$

The following theorem generalizes [9, Theorem 4.7] and [2, Theorem 3.12].

Theorem 3.16.

Let S be a nearly epsilon-strongly G-graded ring. The map $\varphi$ defines a bijection between the set of G-graded ideals of S and the set of G-invariant ideals of ${\oplus }_{e\in G_0}{S}_e.$ The inverse of $\varphi$ is given by $\psi .$

Proof.

Let I be a G-graded ideal of S. Lemma 3.14 implies that $\psi °\varphi (I)=S(I\cap {\oplus }_{e\in G_0}{S}_e)S=I.$ Let J be a G-invariant ideal of S. Notice that, by Proposition 3.5(iii) and Lemma 3.12, $\varphi °\psi (J)=$ $(\mathit{\text{SJS}})\cap {\oplus }_{e\in G_0}{S}_e=J.$ □

3.3 Graded primeness of groupoid graded rings

In this section, we identify necessary and sufficient conditions for graded primeness of a groupoid graded ring.

Definition 3.17.

Let S be a G-graded ring.

1. ${\oplus }_{e\in G_0}{S}_e$ is said to be G-prime if there are no nonzero G-invariant ideals I, J of ${\oplus }_{e\in G_0}{S}_e$ such that $\mathit{\text{IJ}}=\left\{0\right\}$.

2. S is said to be graded prime if there are no nonzero G-graded ideals I, J of S such that $\mathit{\text{IJ}}=\left\{0\right\}$.

The following result generalizes [2, Proposition 3.29].

Theorem 3.18.

Let S be a nearly epsilon-strongly G-graded ring. Then S is graded prime if, and only if, ${\oplus }_{e\in G_0}{S}_e$ is G-prime.

Proof.

We first show the “if” statement. Suppose that ${\oplus }_{e\in G_0}{S}_e$ is G-prime and let I₁, I₂ be nonzero G-graded ideals of S. By Lemma 3.13 and Corollary 3.4, $(I_1\cap {\oplus }_{e\in G_0}{S}_e)$ and $(I_2\cap {\oplus }_{e\in G_0}{S}_e)$ are nonzero G-invariant ideals of ${\oplus }_{e\in G_0}{S}_e.$ Then $\left\{0\right\}\ne (I_1\cap {\oplus }_{e\in G_0}{S}_e)·(I_2\cap {\oplus }_{e\in G_0}{S}_e)\subseteq I_1{I}_2.$

Now, we show the “only if” statement. Suppose that S is graded prime and let J₁, J₂ be nonzero G-invariant ideals of ${\oplus }_{e\in G_0}{S}_e.$ By Proposition 3.5(iii) and Proposition 3.11, $S{J}_{1}S$ and $S{J}_{2}S$ are nonzero G-graded ideals of S. By Corollary 3.15 and our assumption, $S{J}_1·J_{2}S=S{J}_{1}S·S{J}_{2}S\ne \left\{0\right\}.$ Thus, $J_1·J_2\ne \left\{0\right\}$. □

Now, we determine some necessary conditions for graded primeness of a groupoid graded ring.

Lemma 3.19.

Let S be a G-graded ring. Then Sb_tS is a G-graded ideal of S for all $t\in G$ and $b_t\in S_t.$

Proof.

Take $t\in G$ and $b_t\in S_t$. Clearly, $S{b}_{t}S$ is an ideal of S. Notice that ${\oplus }_{g\in G}((S{b}_{t}S)\cap S_g)\subseteq S{b}_{t}S.$ Now, take $k,h\in G,$ $a_k\in S_k$, and $c_h\in S_h.$ If $s(k)\ne r(t)$ or $s(t)\ne r(h),$ then $a_k{b}_t{c}_h=0.$ Otherwise, we have that $$a_k{b}_t{c}_h\in S_k{b}_t{S}_h\subseteq (S{b}_{t}S)\cap S_{\mathit{\text{kth}}}\subseteq {\oplus }_{g\in G}((S{b}_{t}S)\cap S_g).$$

This shows that ${\oplus }_{g\in G}((S{b}_{t}S)\cap S_g)=S{b}_{t}S.$ □

Lemma 3.20.

Let S be a G-graded ring which is s-unital. Suppose that S is graded prime. Let $a_g\in S_g$ and $c_h\in S_h$ be nonzero elements, for some $g,h\in G$. Then there is some $t\in G$ and $b_t\in S_t$ such that $a_g{b}_t{c}_h$ is nonzero.

Proof.

We prove the contrapositive statement. Suppose that $a_g{b}_t{c}_h=0$ for all $t\in G$ and $b_t\in S_t.$ Consider the sets $A:=S{a}_{g}S$ and $C:=S{c}_{h}S$ which, by Lemma 3.19 and the s-unitality of S, are both nonzero G-graded ideals of S. By assumption, we have $\mathit{\text{AC}}=S{a}_g\mathit{\text{SS}}{c}_{h}S=S{a}_{g}S{c}_{h}S=\left\{0\right\}.$ This shows that S is not graded prime. □

Definition 3.21.

Let S be a G-graded ring. An element $e\in G_0^{\prime }$ (see Proposition 3.5(vi)) is said to be a support-hub if for every nonzero $a_g\in S_g$, with $g\in G$, there are $h,k\in G$ such that $s(h)=e,$ r(k) = e, and $a_g{S}_h$ and $S_k{a}_g$ are both nonzero.

Remark 3.22.

Let S be a G-graded ring.

1. Suppose that $e\in G_0^{\prime }$ is a support-hub and that $a_g\in S_g$ is nonzero, for some $g\in G$. Notice that there are $h,k\in G$ as in the following diagram.

   

2. Notice that, if S is a ring which is nearly epsilon-strongly graded by a group G, then the identity element e of G is always a support-hub.

Proposition 3.23.

Let S be a G-graded ring which is s-unital. If S is graded prime, then every $e\in G_0^{\prime }$ is a support-hub.

Proof.

We prove the contrapositive statement. Suppose that there is some $e\in G_0^{\prime }$ which is not a support-hub. Then there are $g\in G$ and a nonzero element $a_g\in S_g$, such that for every $h\in G$ such that $s(h)=e,$ we have that $a_g{S}_h=\left\{0\right\}$ or for every $k\in G$ such that $r(k)=e,$ we have that $S_k{a}_g=\left\{0\right\}.$ Let a_e be a nonzero element of $S_e.$ Using Lemma 3.19 and the fact that S is s-unital, $A:=S{a}_{e}S$ and $B:=S{a}_{g}S$ are nonzero G-graded ideals of S.

Notice that if for every $h\in G$ such that $s(h)=e,$ we have that $a_g{S}_h=\left\{0\right\},$ then $\mathit{\text{BA}}=S{a}_g\mathit{\text{SS}}{a}_{e}S=S{a}_{g}S{a}_{e}S=\left\{0\right\}.$ Moreover, if for every $k\in G$ such that $r(k)=e,$ we have that $S_k{a}_g=\left\{0\right\},$ then $\mathit{\text{AB}}=S{a}_e\mathit{\text{SS}}{a}_{g}S=S{a}_{e}S{a}_{g}S=\left\{0\right\}.$ Therefore, S is not graded prime. □

Proposition 3.24.

Let S be a G-graded ring which is s-unital. The following assertions hold:

1. If G is a connected groupoid, then $G\prime$ is a connected subgroupoid of G.

2. If there is a support-hub in $G_0^{\prime }$, then $G\prime$ is a connected subgroupoid of G.

3. If S is graded prime, then $G\prime$ is a connected subgroupoid of G.

Proof.

1. Suppose that G is connected. Take $e,f\in G_0^{\prime }.$ By assumption, there is $g\in G$ such that s(g) = e and $r(g)=f.$ Since S_e and S_f are nonzero, we must have $g\in G\prime$, and hence $G\prime$ is connected.

2. Suppose that $e\in G_0^{\prime }$ is a support-hub. Take $f_1,f_2\in G_0^{\prime }.$ By the definition of $G\prime ,$ there are nonzero elements $a_{f_1}\in S_{f_1}$ and $b_{f_2}\in S_{f_2}.$ Since e is a support-hub, there is some $k_1\in G$ such that r(k) = e and $S_k{a}_{f_1}\ne \left\{0\right\}.$ In particular, $s(k)=f_1.$ Using again that e is a support-hub, there is some $h\in G$ such that s(h) = e and $b_{f_2}{S}_h\ne \left\{0\right\}.$ Hence, $r(h)=f_2.$ Define $t:=\mathit{\text{hk}}$ and note that $r(t)=f_2$ and $s(t)=f_1.$

3. It follows from Proposition 3.23 and (ii). □

3.4 Primeness of groupoid graded rings

In this section, we will provide necessary and sufficient conditions for primeness of a nearly epsilon-strongly G-graded ring. Furthermore, we will extend [9, Theorem 1.3] to the context of groupoid graded rings.

Proposition 3.25.

Let S be a nearly epsilon-strongly G-graded ring. If S is prime, then ${\oplus }_{g\in G_e^e}{S}_g$ is prime for every $e\in G_0.$

Proof.

We prove the contrapositive statement. Let $e\in G_0.$ Suppose that I and J are nonzero ideals of ${\oplus }_{g\in G_e^e}{S}_g$ such that $\mathit{\text{IJ}}=\left\{0\right\}$. By Proposition 3.5(iii), S is s-unital and hence $A\prime :=\mathit{\text{SIS}}$ and $B\prime :=\mathit{\text{SJS}}$ are nonzero ideals of S. Clearly, $A\prime B\prime =\mathit{\text{SISSJS}}=\mathit{\text{SISJS}}.$ We claim that $\mathit{\text{ISJ}}\subseteq \mathit{\text{IJ}}.$ If we assume that the claim holds, then it follows that $A\prime B\prime =\left\{0\right\}$, and we are done. Now we show the claim. Take $g\in G, c_g\in S_g,$ $a=\sum _{k\in G_e^e}{a}_k\in I,$ and $b=\sum _{t\in G_e^e}{b}_t\in J$. Let $k,t\in G_e^e.$ If $e=s(k)\ne r(g)$ or $s(g)\ne r(t)=e$, then $a_k{c}_g{b}_t=0\in \mathit{\text{IJ}}.$ Otherwise, $g\in G_e^e,$ and then, since I and J are ideals of ${\oplus }_{g\in G_e^e}{S}_g$, we get that $a{c}_{g}b\in \mathit{\text{IJ}}.$ Thus, $\mathit{\text{ISJ}}\subseteq \mathit{\text{IJ}}$. □

Remark 3.26.

Let S be a nearly epsilon-strongly G-graded ring.

1. By Propositions 3.25 and 3.24(iii), if S is prime, then ${\oplus }_{g\in G_e^e}{S}_g$ is prime for every $e\in G_0$ and $G\prime$ is connected. The converse, however, need not hold as shown by Example 4.22.

2. Recall that, by Lemma 2.4, ${\pi }_{G_e^e}:S\to S_{G_e^e}$ is defined by ${\pi }_{G_e^e}(\sum _{g\in G}{c}_g):=\sum _{h\in G_e^e}{c}_h$ for every $e\in G_0$.

The next result partially generalizes [9, Lemma 2.19].

Lemma 3.27.

Let S be a nearly epsilon-strongly G-graded ring and let I be a nonzero ideal of S. If $e\in G_0^{\prime }$ is a support-hub, then ${\pi }_{G_e^e}(I)$ is a nonzero ideal of ${\oplus }_{g\in G_e^e}{S}_g.$

Proof.

Suppose that $e\in G_0^{\prime }$ is a support-hub. By Lemma 2.4, ${\pi }_{G_e^e}(I)$ is an ideal of ${\oplus }_{g\in G_e^e}{S}_g.$ We claim that ${\pi }_{G_e^e}(I)\ne \left\{0\right\}.$ Let $d=d_{g_1}+d_{g_2}+\cdots +d_{g_n}\in I$ be an element where all the homogeneous coefficients are nonzero and the $g_i^{\prime }s$ are distinct. By Corollary 3.4, there is some nonzero $c_{g_1^{-1}}\in S_{g_1^{-1}}$ such that $d_{g_1}{c}_{g_1^{-1}}$ is nonzero and contained in $S_{r(g_1)}$.

Notice that $d{c}_{g_1^{-1}}$ is nonzero and contained in I. Thus, without loss of generality, we may assume that $g_1\in G_0^{\prime }$. Since e is a support-hub, there is an element $k\in G$ such that r(k) = e and $S_k{d}_{g_1}$ is nonzero. In particular, there is an element $b_k\in S_k$ such that $b_k{d}_{g_1}$ is nonzero. Using again that e is a support-hub, there is an element $h\in G$ such that s(h) = e and $(b_k{d}_{g_1})S_h$ is nonzero. Therefore, there is an element $b_h\in S_h$ such that $(b_k{d}_{g_1})b_h$ is nonzero. Hence, $b_{k}d{b}_h\in I$ and $${\pi }_{G_e^e}(b_{k}d{b}_h)=\sum _{\begin{array}{c}s(k)=r(g_i)\\ s(g_i)=r(h)\end{array}}{b}_k{d}_{g_i}{b}_h=\sum _{\begin{array}{c}{g}_1=r(g_i)\\ s(g_i)=g_1\end{array}}{b}_k{d}_{g_i}{b}_h.$$

Notice that $b_k{d}_{g_i}{b}_h\in S_{\mathit{\text{kh}}}\setminus \left\{0\right\}$ if, and only if, $g_i=g_1.$ Thus, $0\ne {\pi }_{G_e^e}(b_{k}d{b}_h)\in {\pi }_{G_e^e}(I).$ □

Theorem 3.28.

Let S be a nearly epsilon-strongly G-graded ring. If there is some $e\in G_0^{\prime }$ such that e is a support-hub and ${\oplus }_{g\in G_e^e}{S}_g$ is prime, then S is prime.

Proof.

Suppose that $e\in G_0^{\prime }$ is a support-hub and that ${\oplus }_{g\in G_e^e}{S}_g$ is prime. Let I and J be nonzero ideals of S. By Lemma 3.27, ${\pi }_{G_e^e}(I)$ and ${\pi }_{G_e^e}(J)$ are nonzero ideals of ${\oplus }_{g\in G_e^e}{S}_g,$ and hence, by assumption, ${\pi }_{G_e^e}(I){\pi }_{G_e^e}(J)\ne \left\{0\right\}.$

We claim that ${\pi }_{G_e^e}(I)\subseteq I.$ Let $d=\sum _{t\in G}{d}_t\in I$ and consider the finite set $F:=\{t\in \text{Supp}(d):r(t)=e\}.$ If $F=\varnothing$, then ${\pi }_{G_e^e}(d)=0\in I$. Now, suppose that $F\ne \varnothing$ and take $t\in F$. Using that $S_t=S_t{S}_{t^{-1}}{S}_t,$ there are $n_t\in {\mathbb{Z}}^{+}$, $a_1^t,\dots ,a_{n_t}^t\in S_t{S}_{t^{-1}}\subseteq S_{r(t)}=S_e$ and $b_1^t,\dots ,b_{n_t}^t\in S_t$ such that $d_t={\sum }_{i=1}^{n_t}{a}_i^t{b}_i^t.$

By Proposition 2.6, using that S_e is s-unital, there is some $u_e\in S_e$ such that $u_e{a}_i^t=a_i^t$ for all $t\in F$ and all $i\in \{1,\dots ,n_t\}.$ Thus, $d_t=u_e{d}_t$ for all $t\in F$ and $u_e\sum _{t\in G}{d}_t=\sum _{t\in F}{u}_e{d}_t=\sum _{t\in F}{d}_t.$

Using a similar argument, there is some $v_e\in S_e$ such that $d_t=d_t{v}_e$ for every $t\in \text{Supp}(d)$ such that s(t) = e. Therefore, $$I\ni \left(u_e\sum _{t\in G}{d}_t\right)v_e=\left(\sum _{t\in F}{d}_t\right)v_e=\sum _{t\in G_e^e}{d}_t{v}_e=\sum _{t\in G_e^e}{d}_t={\pi }_{G_e^e}(d).$$

Analogously, ${\pi }_{G_e^e}(J)\subseteq J.$ Thus, $\left\{0\right\}\ne {\pi }_{G_e^e}(I){\pi }_{G_e^e}(J)\subseteq \mathit{\text{IJ}}$ and S is prime. □

Remark 3.29.

The assumption on the existence of a support-hub in Theorem 3.28 cannot be dropped. Indeed, consider the groupoid $G=\{e,f\}=G_0$ and the groupoid ring $S:=\mathbb{C}[G]$. Then $G_e^e=\left\{e\right\}$ and $G_f^f=\left\{f\right\}$. Furthermore, $S_e\cong \mathbb{C}$ and $S_f\cong \mathbb{C}$ are both prime. Nevertheless, S is not prime.

The next example shows that the existence of a support-hub in a connected grading groupoid is not enough to guarantee (graded) primeness of the graded ring.

Example 3.30.

Let $G:=\{f_1,f_2,f_3,g,h,g^{-1},h^{-1},h{g}^{-1},g{h}^{-1}\}$ be a groupoid with $G_0=\{f_1,f_2,f_3\}$ and depicted as follows:

Define S as the ring of matrices over $\mathbb{Z}$ of the form $$\left(\begin{array}{cccc}{a}_{11}& a_{12}& 0& 0\\ a_{21}& a_{22}& 0& 0\\ 0& 0& a_{33}& a_{34}\\ 0& 0& a_{43}& a_{44}\end{array}\right).$$

Denote by ${\left\{e_{\mathit{\text{ij}}}\right\}}_{i,j},$ the standard matrix units and define: $$S_g:=\mathbb{Z}{e}_{12},S_{g^{-1}}:=\mathbb{Z}{e}_{21},S_h:=\mathbb{Z}{e}_{43},S_{h^{-1}}:=\mathbb{Z}{e}_{34},$$ $$S_{f_1}:=\mathbb{Z}{e}_{11},S_{f_3}:=\mathbb{Z}{e}_{44},S_{f_2}:=\{{\lambda }_1{e}_{22}+{\lambda }_2{e}_{33}:{\lambda }_1,{\lambda }_2\in \mathbb{Z}\},$$and $S_l:=\left\{0\right\},$ otherwise. It is not difficult to verify that this G-grading is nearly epsilon-strong. Notice that $S={\oplus }_{l\in G}{S}_l$ and that $f_2\in G_0^{\prime }$ is a support-hub. However, observe that $e_{12}\in S_g$ and $e_{43}\in S_h$ are nonzero elements and that there is no element $a_l\in S_l$ such that $l\in G\prime$ and $e_{12}{a}_l{e}_{43}\ne 0.$ Therefore, by Lemma 3.20, S is not graded prime.

Now, we prove our main result.

Proof of Theorem 1.1.

It follows from Proposition 3.25 and by the definition of primeness that (i) $\Rightarrow$ (iv) $\Rightarrow$ (v). By Proposition 3.5(iii), S is s-unital and Proposition 3.23 implies (iv) $\Rightarrow$ (vi) $\Rightarrow$ (vii) and (v) $\Rightarrow$ (vii). By Theorem 3.28, (vii) $\Rightarrow$ (i). Finally, note that by Theorem 3.18, (ii) is equivalent to (iv), and (iii) is equivalent to (v). □

Remark 3.31.

In [14], Munn investigates primeness of rings graded by inverse semigroups. He shows (see [14, Theorem 4.1]) that if S is a so-called 0-bisimple inverse semigroup, R is a faithful restricted S-graded ring, and R_G is prime for some nonzero maximal subgroup G of S, then R is prime. We point out that Munn’s theorem can potentially be used to prove e.g. (vi) $\Rightarrow$(i) in Theorem 1.1. Indeed, we may associate a natural inverse semigroup $S(G):=G\cup \left\{z\right\}$ with the groupoid G and view any G-graded ring as an S(G)-graded ring (see e.g., [10, Section 4.3]). It is easy to come up with examples of prime nearly-epsilon strongly G-graded rings such that the corresponding S(G)-gradings fail to satisfy the requirements in Munn’s theorem. However, given a prime nearly epsilon-strongly G-graded ring R, it is not clear to the authors whether one can always find a subgroupoid H of G, contained in $\text{Supp}(R)$, such that S(H) and its grading on R do in fact satisfy the requirements in Munn’s theorem.

We recall that Passman [22] provided a characterization of prime unital strongly group graded rings. That result was generalized in [9, Theorem 1.3] to nearly epsilon-strongly group graded rings.

Theorem 3.32

([9, Theorem 1.3]). Let G be a group and let S be a nearly epsilon-strongly G-graded ring. The following statements are equivalent:

1. S is not prime;

2. There exist:

   1. subgroups $N◁H\subseteq G$,

   2. an H-invariant ideal I of S_e such that $I^{g}I=\left\{0\right\}$ for all $g\in G\setminus H,$ and

   3. nonzero ideals $\tilde{A},\tilde{B}$ of S_N such that $\tilde{A},\tilde{B}\subseteq I{S}_N$ and $\tilde{A}{S}_H\tilde{B}=\left\{0\right\}.$

3. There exist:

   1. subgroups $N◁H\subseteq G$ with N finite,

   2. an H-invariant ideal I of S_e such that $I^{g}I=\left\{0\right\}$ for all $g\in G\setminus H,$ and

   3. nonzero ideals $\tilde{A},\tilde{B}$ of S_N such that $\tilde{A},\tilde{B}\subseteq I{S}_N$ and $\tilde{A}{S}_H\tilde{B}=\left\{0\right\}.$

4. There exist:

   1. subgroups $N◁H\subseteq G$ with N finite,

   2. an H-invariant ideal I of S_e such that $I^{g}I=\left\{0\right\}$ for all $g\in G\setminus H,$ and

   3. nonzero H-invariant ideals $\tilde{A},\tilde{B}$ of S_N such that $\tilde{A},\tilde{B}\subseteq I{S}_N$ and $\tilde{A}{S}_H\tilde{B}=\left\{0\right\}.$

5. There exist:

   1. subgroups $N◁H\subseteq G$ with N finite,

   2. an H-invariant ideal I of S_e such that $I^{g}I=\left\{0\right\}$ for all $g\in G\setminus H,$ and

   3. nonzero H/N-invariant ideals $\tilde{A},\tilde{B}$ of S_N such that $\tilde{A},\tilde{B}\subseteq I{S}_N$ and $\tilde{A}\tilde{B}=\left\{0\right\}.$

Remark 3.33.

Note that, in Theorem 1.1, ${\oplus }_{g\in G_e^e}{S}_g$ is nearly epsilon-strongly graded by the group $G_e^e$. Hence, one can use Theorem 3.32 to decide whether ${\oplus }_{g\in G_e^e}{S}_g$ is prime.

The following Theorem generalizes [9, Theorem 1.4].

Theorem 3.34.

Let S be a nearly epsilon-strongly G-graded ring. Suppose that there is some $e\in G_0^{\prime }$ such that $G_e^e$ is torsion-free. Then S is prime if, and only if, S_e is $G_e^e$-prime and ${\oplus }_{e\in G_0}{S}_e$ is G-prime.

Proof.

It follows from Theorem 1.1 and [9, Theorem 1.4]. □

4 Applications to partial skew groupoid rings

In this section, we will apply our main results on primeness for nearly epsilon-strongly groupoid graded rings to partial skew groupoid rings, (global) skew groupoid rings, and groupoid rings. In particular, we will characterize prime partial skew groupoid rings induced by partial actions of group-type (cf. [4]). Furthermore, we will generalize [9, Theorem 12.4] and [9, Theorem 13.7].

Throughout this section, unless stated otherwise, G denotes an arbitrary groupoid and A denotes an arbitrary ring.

4.1 Partial skew groupoid rings

Definition 4.1.

A partial action of a groupoid G on a ring A is a family of pairs $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ satisfying:

1. For each $g\in G,$ $A_{r(g)}$ is an ideal of A, A_g is an ideal of $A_{r(g)}$, and ${\sigma }_g:A_{g^{-1}}\to A_g$ is a ring isomorphism,

2. ${\sigma }_e={\text{id}}_{A_e}$, for every $e\in G_0$,

3. ${\sigma }_h^{-1}(A_{g^{-1}}\cap A_h)\subseteq A_{{(\mathit{\text{gh}})}^{-1}}$, whenever $(g,h)\in G^2$,

4. ${\sigma }_g({\sigma }_h(x))={\sigma }_{\mathit{\text{gh}}}(x)$, for all $x\in {\sigma }_h^{-1}(A_{g^{-1}}\cap A_h)$ and $(g,h)\in G^2$.

Definition 4.2.

Given a partial action σ of a groupoid G on a ring A one may define the partial skew groupoid ring $A{⋊}_{\sigma }G$ as the set of all formal sums of the form $\sum _{g\in G}{a}_g{\delta }_g$, where $a_g\in A_g$ is zero for all but finitely many $g\in G$, and with addition defined point-wise and multiplication given by $$a_g{\delta }_g·b_h{\delta }_h=\{\begin{array}{cc}{\sigma }_g({\sigma }_{g^{-1}}(a_g)b_h){\delta }_{\mathit{\text{gh}}},\hfill & \text{if}(g,h)\in G^2,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$

Remark 4.3.

(a) Throughout this section, unless stated otherwise, will assume that $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ is an arbitrary partial action of G on A, that A_g is an s-unital ring, for every $g\in G$, and that $A={\oplus }_{e\in G_0}{A}_e$. As a consequence, $A{⋊}_{\sigma }G$ will always be an associative ring (see [2, Remark 2.7 (ii)]), and there will exist a ring isomorphism (cf. [2, Lemma 3.6 (i)]) $\psi :A\to {\oplus }_{e\in G_0}{A}_e{\delta }_e$ defined by (1) $$\psi \mathrm{}\left(\sum _{e\in G_0}{a}_e\right):=\sum _{e\in G_0}{a}_e{\delta }_e.$$(1)

(b) Under the above assumptions, by [2, Lemma 3.2], A_g is an ideal of A, for every $g\in G.$

(c) It is readily verified that any partial skew groupoid ring $S:=A{⋊}_{\sigma }G$ carries a natural G-grading defined by letting $S_g:=A_g{\delta }_g$, for every $g\in G$.

The following result generalizes [9, Proposition 13.1].

Proposition 4.4.

The partial skew groupoid ring $A{⋊}_{\sigma }G$ is a nearly epsilon-strongly G-graded ring.

Proof.

Let $g\in G$. Using that $A_{g^{-1}}$ is s-unital, and hence idempotent, we get that $$(A_g{\delta }_g)(A_{g^{-1}}{\delta }_{g^{-1}})={\sigma }_g({\sigma }_{g^{-1}}(A_g)A_{g^{-1}}){\delta }_{r(g)}={\sigma }_g(A_{g^{-1}}{A}_{g^{-1}}){\delta }_{r(g)}={\sigma }_g(A_{g^{-1}}){\delta }_{r(g)}=A_g{\delta }_{r(g)}.$$

Now, using that A_g is s-unital we get that $A_g{\delta }_{r(g)}$ is s-unital, and that A_g is idempotent. Hence, $$(A_g{\delta }_g)(A_{g^{-1}}{\delta }_{g^{-1}})(A_g{\delta }_g)=A_g{\delta }_{r(g)}{A}_g{\delta }_g=A_g^2{\delta }_g=A_g{\delta }_g.$$

This shows that $A{⋊}_{\sigma }G$ is nearly epsilon-strongly G-graded. □

Remark 4.5.

By Propositions 3.5 and 4.4, the partial skew groupoid ring $A{⋊}_{\sigma }G$ is s-unital.

Definition 4.6.

Let G be a groupoid, let A be a ring and let $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ be a partial action of G on A.

1. Let H be a subgroupoid of G. An ideal I of A is said to be H-invariant if ${\sigma }_g(I\cap A_{g^{-1}})\subseteq I$ for every $g\in H$.

2. A is said to be G-prime if there are no nonzero G-invariant ideals I, J of A such that $\mathit{\text{IJ}}=\left\{0\right\}$.

The next result generalizes [9, Remark 13.4] from the group setting.

Proposition 4.7.

Let I be an ideal of A. Then I is G-invariant in the sense of Definition 4.6 if, and only if, $\psi (I)$ is a G-invariant ideal of ${\oplus }_{e\in G_0}{A}_e{\delta }_e$ in the sense of Definition 3.8. In particular, A is G-prime if, and only if, ${\oplus }_{e\in G_0}{A}_e{\delta }_e$ is G-prime.

Proof.

Suppose that I is an ideal of A. Let $g\in G.$ By Remark 4.3(b) and the s-unitality of A_g, we get that $A_{g}I=I\cap A_g=I{A}_g$ and $I{A}_g=A_{g}I{A}_g=A_{g}I$. Furthermore, $\psi (I)={\oplus }_{e\in G_0}(I{A}_e){\delta }_e.$ Notice that $A_g=A_{r(g)}{A}_g$. We get that $$\begin{array}{cc}\psi {(I)}^g\hfill & =A_{g^{-1}}{\delta }_{g^{-1}}·\psi (I)·A_g{\delta }_g=A_{g^{-1}}{\delta }_{g^{-1}}·(I{A}_{r(g)})A_g{\delta }_g=A_{g^{-1}}{\delta }_{g^{-1}}·I{A}_g{\delta }_g\hfill \\ \hfill & ={\sigma }_{g^{-1}}({\sigma }_g(A_{g^{-1}})I{A}_g){\delta }_{s(g)}={\sigma }_{g^{-1}}(A_{g}I{A}_g){\delta }_{s(g)}.\hfill \end{array}$$

Therefore, $$\psi {(I)}^g\subseteq \psi (I)\iff {\sigma }_{g^{-1}}(A_{g}I{A}_g){\delta }_{s(g)}\subseteq \psi (I)\iff {\sigma }_{g^{-1}}(A_{g}I{A}_g)\subseteq I\iff {\sigma }_{g^{-1}}(I\cap A_g)\subseteq I.$$

□

Remark 4.8.

1. Recall that, with the natural G-grading on $A{⋊}_{\sigma }G$, an element $e\in G_0^{\prime }$ is a support-hub if for every nonzero element $a_g{\delta }_g$, with $g\in G$, there are $h,k\in G$ such that $s(h)=e,$ r(k) = e and both $a_g{\delta }_g{A}_h{\delta }_h$ and $A_k{\delta }_k{a}_g{\delta }_g$ are nonzero.

2. For $e\in G_0$, denote by ${\sigma }^e:={(A_h,{\sigma }_h)}_{h\in G_e^e}$ the partial action of the isotropy group $G_e^e$ on the ring A_e, obtained by restricting σ. The associated partial skew group ring is denoted by $A_e{⋊}_{{\sigma }^e}{G}_e^e$.

Theorem 4.9.

Let $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ be a partial action of G on A such that A_g is s-unital for every $g\in G$ and $A={\oplus }_{e\in G_0}{A}_e$. Then, the following statements are equivalent:

1. The partial skew groupoid ring $A{⋊}_{\sigma }G$ is prime;

2. A is G-prime and, for every $e\in G_0^{\prime },$ $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime;

3. A is G-prime and, for some $e\in G_0^{\prime }, A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime;

4. $A{⋊}_{\sigma }G$ is graded prime and, for every $e\in G_0^{\prime },$ $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime;

5. $A{⋊}_{\sigma }G$ is graded prime and, for some $e\in G_0^{\prime }, A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime;

6. For every $e\in G_0^{\prime },$ e is a support-hub and $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime;

7. For some $e\in G_0^{\prime },$ e is a support-hub and $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime.

Proof.

It follows from Proposition 4.4, Theorem 1.1, and Proposition 4.7. □

We recall the following result from [9]. In that paper, the authors say that a partial skew group ring is s-unital if it is defined by a partial group action on s-unital ideals.

Theorem 4.10

([9, Theorem 13.7]). Let G be a group and let $A{⋊}_{\sigma }G$ be an s-unital partial skew group ring. Then, $A{⋊}_{\sigma }G$ is not prime if, and only if, there are:

1. subgroups $N◁H\subseteq G$ with N finite,

2. an ideal I of A such that

   • ${\sigma }_h(I{A}_{h^{-1}})=I{A}_h$ for every $h\in H,$

   • $I{A}_g·{\sigma }_g(I{A}_{g^{-1}})=\left\{0\right\}$ for every $g\in G\setminus H,$ and

3. (iii) nonzero ideals $\tilde{B},\tilde{D}$ of $A{⋊}_{\sigma }N$ such that $\tilde{B},\tilde{D}\subseteq I{\delta }_e(A{⋊}_{\sigma }N)$ and $\tilde{B}·A_h{\delta }_h·\tilde{D}=\left\{0\right\}$ for every $h\in H.$

Remark 4.11.

Note that, in Theorem 4.9, $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is an s-unital partial skew group ring. Thus, one can apply Theorem 4.10 to determine whether $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime.

Definition 4.12

([4, Remark 3.4]). A partial action $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ of a connected groupoid G on a ring A is said to be of group-type if there exist an element $e\in G_0$ and a family of morphisms ${\left\{h_f\right\}}_{f\in G_0}$ in G such that $h_f:e\to f,$ $h_e=e,$ $A_{h_f^{-1}}=A_e$ and $A_{h_f}=A_f,$ for every $f\in G_0.$

Remark 4.13.

1. If a partial action σ is of group-type (and hence G is connected), then every element of G₀ can take the role of e in the above definition (see [4, Remark 3.4]).

2. By [4, Lemma 3.1], every global action by a connected groupoid is of group-type. The converse does not hold. For an example of a non-global partial action of group-type, we refer the reader to [4, Example 3.5].

Lemma 4.14.

Let $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ be a partial action of G on A such that A_g is s-unital for every $g\in G$ and $A={\oplus }_{e\in G_0}{A}_e$. Furthermore, let $e\in G_0^{\prime }$ and consider the following statements:

1. σ is of group-type (and G is connected);

2. For every nonzero element $a_g{\delta }_g\in A{⋊}_{\sigma }G$ there is some $k\in G$ such that $s(k)=r(g),$ r(k) = e and $a_g\in A_{k^{-1}};$

3. For every nonzero element $a_g{\delta }_g\in A{⋊}_{\sigma }G$ there is some $k\in G$ such that $s(k)=r(g),$ r(k) = e and $A_{k^{-1}}{a}_g\ne \left\{0\right\};$

4. e is a support-hub.

Then, (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) $\iff$ (iv).

Proof.

(i) $\Rightarrow$ (ii) Suppose that (i) holds. Let $g\in G$ and $a_g{\delta }_g\ne 0.$ By Remark 4.13(a), since σ is of group-type, there is a morphism $h_{r(g)}:e\to r(g)$ such that $A_{h_{r(g)}^{-1}}=A_e$ and $A_{h_{r(g)}}=A_{r(g)}.$ Note that $a_g\in A_g\subseteq A_{r(g)}=A_{h_{r(g)}}.$ Define $k:=h_{r(g)}^{-1}$ and the proof is done.

(ii) $\Rightarrow$ (iii) Suppose that (ii) holds. Let $g\in G$ and $a_g{\delta }_g\ne 0.$ By assumption, there is some $k\in G$ such that $s(k)=r(g),$ r(k) = e and $a_g\in A_{k^{-1}}.$ Since $A_{k^{-1}}$ is s-unital, we get $A_{k^{-1}}{a}_g\ne \left\{0\right\}.$

(iii) $\Rightarrow$ (iv) Suppose that (iii) holds. Let $g\in G$ and $a_g{\delta }_g\ne 0.$ By assumption, there is some $k\in G$ such that $s(k)=r(g)$ and r(k) = e and $A_{k^{-1}}{a}_g\ne \left\{0\right\}.$ Hence, there is some $d_{k^{-1}}\in A_{k^{-1}}$ such that $0\ne d_{k^{-1}}{a}_g\in A_{k^{-1}}\cap A_g.$ Let $u\in A_{g^{-1}}$ be an s-unit for ${\sigma }_{g^{-1}}(d_{k^{-1}}{a}_g)$ and let $w\in A_{k^{-1}}$ be an s-unit for $d_{k^{-1}}{a}_g.$ Note that $$b:=(({\sigma }_k(w){\delta }_k)(d_{k^{-1}}{\delta }_{s(k)}))(a_g{\delta }_g)((u{\delta }_{g^{-1}})(w{\delta }_{k^{-1}}))\in A_k{\delta }_k(a_g{\delta }_g)A_{g^{-1}{k}^{-1}}{\delta }_{g^{-1}{k}^{-1}},$$and, $$\begin{array}{cc}b\hfill & =({\sigma }_k(w){\delta }_k)(d_{k^{-1}}{a}_g{\delta }_g)(u{\delta }_{g^{-1}})(w{\delta }_{k^{-1}})=({\sigma }_k(w){\delta }_k)({\sigma }_g({\sigma }_{g^{-1}}(d_{k^{-1}}{a}_g)u){\delta }_{g{g}^{-1}})(w{\delta }_{k^{-1}})\hfill \\ \hfill & =({\sigma }_k(w){\delta }_k)(d_{k^{-1}}{a}_g{\delta }_{r(g)})(w{\delta }_{k^{-1}})=({\sigma }_k(w){\delta }_k)(d_{k^{-1}}{a}_g{\delta }_{k^{-1}})={\sigma }_k({\sigma }_{k^{-1}}({\sigma }_k(w))d_{k^{-1}}{a}_g){\delta }_{k{k}^{-1}}\hfill \\ \hfill & ={\sigma }_k(d_{k^{-1}}{a}_g){\delta }_e\ne 0.\hfill \end{array}$$

Define $h:=g^{-1}{k}^{-1}$ and note that $s(g^{-1}{k}^{-1})=e$. Moreover, we have that $A_k{\delta }_k{a}_g{\delta }_g$ and $a_g{\delta }_g{A}_h{\delta }_h$ are both nonzero. This shows that e is a support-hub.

(iv) $\Rightarrow$ (iii) Suppose that (iv) holds. Let $g\in G$ and $a_g{\delta }_g\ne 0.$ By assumption, there is some $k\in G$ such that r(k) = e and $A_k{\delta }_k{a}_g{\delta }_g$ is nonzero. Note that $A_k{\delta }_k{a}_g{\delta }_g={\sigma }_k(A_{k^{-1}}{a}_g){\delta }_{\mathit{\text{kg}}}.$ Therefore, $A_{k^{-1}}{a}_g\ne \left\{0\right\}.$ □

Theorem 4.15.

Let $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ be a partial action of a connected groupoid G on A such that A_g is s-unital for every $g\in G,$ $A={\oplus }_{e\in G_0}{A}_e$ and σ is of group-type. Then the partial skew groupoid ring $A{⋊}_{\sigma }G$ is prime if, and only if, there is some $e\in G_0^{\prime }$ such that $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime.

Proof.

It follows from Lemma 4.14 and Theorem 4.9. □

Now, we will make use of the example from [4, Example 3.5] of a non-global partial action of a connected groupoid on a ring, of group-type, and apply Theorem 4.15 to it.

Example 4.16.

Let $G=\{e,f,g,h,l,m,l^{-1},m^{-1}\}$ be the groupoid with $G_0=\{e,f\}$ and the following composition rules: $$g^2=e,h^2=f,\mathit{\text{lg}}=m=\mathit{\text{hl}},g\in G_e^e,h\in G_f^f\text{and}l,m:e\to f.$$

We present in the following diagram the structure of G:

Let $\mathbb{C}$ be the field of complex numbers and let $A=\mathbb{C}{e}_1\oplus \mathbb{C}{e}_2\oplus \mathbb{C}{e}_3\oplus \mathbb{C}{e}_4,$ where $e_i{e}_j={\delta }_{i,j}{e}_i$ and $e_1+\cdots +e_4=1.$ We define the partial action ${(A_t,{\sigma }_t)}_{t\in G}$ of G on A as follows: $$\begin{array}{cc}\hfill & A_e=\mathbb{C}{e}_1\oplus \mathbb{C}{e}_2=A_{l^{-1}},A_f=\mathbb{C}{e}_3\oplus \mathbb{C}{e}_4=A_l,\hfill \\ \hfill & A_g=\mathbb{C}{e}_1=A_{g^{-1}}=A_{m^{-1}},A_m=A_h=\mathbb{C}{e}_3=A_{h^{-1}},\hfill \end{array}$$and $$\begin{array}{cc}\hfill & {\sigma }_e={\text{id}}_{A_e},{\sigma }_f={\text{id}}_{A_f},{\sigma }_g:a{e}_1\mapsto \overline{a}{e}_1,{\sigma }_h:a{e}_3\mapsto \overline{a}{e}_3,{\sigma }_m:a{e}_1\mapsto \overline{a}{e}_3,\hfill \\ \hfill & {\sigma }_{m^{-1}}:a{e}_3\mapsto \overline{a}{e}_1,{\sigma }_l:a{e}_1+b{e}_2\mapsto a{e}_3+b{e}_4,{\sigma }_{l^{-1}}:a{e}_3+b{e}_4\mapsto a{e}_1+b{e}_2,\hfill \end{array}$$ where $\overline{a}$ denotes the complex conjugate of a, for all $a\in \mathbb{C}.$ By choosing $h_e:=e$ and $h_f:=l,$ we notice that σ is of group-type (cf. Definition 4.12).

Now, we describe the group partial action ${\sigma }^e={(A_t,{\sigma }_t)}_{t\in G_e^e}$ of $G_e^e$ on $A_e.$ Note that $$G_e^e=\{e,g\},{\sigma }_e={\text{id}}_{A_e}\text{and}{\sigma }_g:a{e}_1\mapsto \overline{a}{e}_1.$$

We claim that A_e is not $G_e^e$-prime. Let $I:=\mathbb{C}{e}_1\oplus \left\{0\right\}{e}_2$ and $J:=\left\{0\right\}{e}_1\oplus \mathbb{C}{e}_2.$ Note that I and J are nonzero $G_e^e$-invariant ideals of A_e and $\mathit{\text{IJ}}=\left\{0\right\}.$ By Theorem 4.9, $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is not prime. An analogous argument shows that $A_f{⋊}_{{\sigma }^f}{G}_f^f$ is not prime. Hence, Theorem 4.15 implies that $A{⋊}_{\sigma }G$ is not prime.

The following result generalizes [9, Theorem 13.5] from the group setting.

Theorem 4.17.

Let $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ be a partial action of G on A such that A_g is s-unital for every $g\in G,$ and $A={\oplus }_{e\in G_0}{A}_e.$ Furthermore, suppose that there is some $e\in G_0^{\prime }$ such that $G_e^e$ is torsion-free. Then $A{⋊}_{\sigma }G$ is prime if, and only if, A_e is $G_e^e$-prime and A is G-prime.

Proof.

It follows from Propositions 4.4, 4.7, and Theorem 3.34. □

4.2 Skew groupoid rings

The partial action $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ of G on A is said to be global if $A_g=A_{r(g)}$ for every $g\in G$. In that case, the corresponding partial skew groupoid ring $A{⋊}_{\sigma }G$ is said to be a skew groupoid ring (see e.g., [20, 21]).

Remark 4.18.

Let $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ be a global action of G on A.

1. For $g\in G, {\sigma }_g:A_{s(g)}\to A_{r(g)}$ is a ring isomorphism.

2. Note that ${\sigma }_g°{\sigma }_h={\sigma }_{\mathit{\text{gh}}}$, whenever $(g,h)\in G^{(2)}$.

3. The multiplication rule on the skew groupoid ring is induced by the following somewhat simplified rule compared to the partial case: $$(a{\delta }_g)(b{\delta }_h):=\{\begin{array}{cc}a{\sigma }_g(b){\delta }_{\mathit{\text{gh}}},\hfill & \text{if}(g,h)\in G^{(2)}\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}$$

for $g,h\in G$ and $a\in A_{r(g)}, b\in A_{r(h)}$.

Theorem 4.19.

Let $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ be a global action of G on A such that A_e is s-unital for every $e\in G_0$, and let $A={\oplus }_{e\in G_0}{A}_e$. Suppose that the groupoid G is connected. Then, the skew groupoid ring $A{⋊}_{\sigma }G$ is prime if, and only if, there is some $e\in G_0^{\prime }$ such that $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime.

Proof.

It follows from Remark 4.13(b) and Theorem 4.15. □

Proposition 4.20.

Let $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ be a global action of G on A such that A_e is s-unital for every $e\in G_0$, and let $A={\oplus }_{e\in G_0}{A}_e$. The following statements are equivalent:

1. $G\prime$ is connected;

2. For every $e\in G_0^{\prime },$ e is a support-hub;

3. For some $e\in G_0^{\prime },$ e is a support-hub.

Proof.

Obviously, (ii) $\Rightarrow$ (iii).

(iii) $\Rightarrow$ (i) This follows from Proposition 3.24(ii).

(i) $\Rightarrow$ (ii) Suppose that $G\prime$ is connected. Take $e\in G_0^{\prime }, g\in G$, and let $a_g{\delta }_g\in A{⋊}_{\sigma }G$ be a nonzero element. Then $g\in G\prime$ and, by assumption, there is some $k\in G\prime$ such that $s(k)=r(g)$ and $r(k)=e.$ Notice that $a_g\in A_{r(g)}=A_{s(k)}=A_{k^{-1}}.$ By Lemma 4.14 (ii) $\Rightarrow$ (iv), e is a support-hub. □

Below, we summarize our findings for skew groupoid rings.

Theorem 4.21.

Let $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ be a global action of G on A such that A_e is s-unital for every $e\in G_0$, and let $A={\oplus }_{e\in G_0}{A}_e$. Then, the following statements are equivalent:

1. The skew groupoid ring $A{⋊}_{\sigma }G$ is prime;

2. A is G-prime, and for every $e\in G_0^{\prime },$ $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime;

3. A is G-prime, and for some $e\in G_0^{\prime },$ $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime;

4. $A{⋊}_{\sigma }G$ is graded prime, and for every $e\in G_0^{\prime },$ $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime;

5. $A{⋊}_{\sigma }G$ is graded prime, and for some $e\in G_0^{\prime },$ $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime;

6. $G\prime$ is connected, and for every $e\in G_0^{\prime },$ $A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime;

7. $G\prime$ is connected, and for some $e\in G_0^{\prime }, A_e{⋊}_{{\sigma }^e}{G}_e^e$ is prime.

Proof.

It follows from Theorem 4.9 and Proposition 4.20. □

The following example shows that Theorem 4.21 does not generalize to partial skew groupoid rings.

Example 4.22.

Let $G=\{e,f,g,g^{-1}\}$ be a groupoid such that $G_0=\{e,f\},$ s(g) = f and r(g) = e as follows:

Let $A:=\mathbb{Z}\oplus \mathbb{Z}.$ Now, we define a partial action $\sigma :={(A_g,{\sigma }_g)}_{g\in G}$ of G on A:

• $A_e:=\mathbb{Z}\oplus \left\{0\right\}$ and $A_f:=\left\{0\right\}\oplus \mathbb{Z};$

• $A_g:=A_{g^{-1}}:=\left\{0\right\};$

• ${\sigma }_e:={\text{id}}_{\mathbb{Z}\oplus \left\{0\right\}},$ ${\sigma }_f:={\text{id}}_{\left\{0\right\}\oplus \mathbb{Z}}$ and ${\sigma }_g:={\sigma }_{g^{-1}}:={\text{id}}_{\left\{0\right\}}.$

Notice that $A_e{⋊}_{{\sigma }^e}{G}_e^e\cong \mathbb{Z}$ and $A_f{⋊}_{{\sigma }^f}{G}_f^f\cong \mathbb{Z}$ are prime rings. Observe that $G=G\prime$, and that G is connected. However, there are no $t\in G$ and $c_t{\delta }_t\in A{⋊}_{\sigma }G$ such that ${\delta }_e(c_t{\delta }_t){\delta }_f\ne 0$. Hence, by Lemma 3.20, $A{⋊}_{\sigma }G$ is not graded prime.

Corollary 4.23.

Let $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ be a global action of G on A such that A_e is s-unital for every $e\in G_0$, and let $A={\oplus }_{e\in G_0}{A}_e$. Suppose that there is some $e\in G_0^{\prime }$ such that $G_e^e$ is torsion-free. Then, the skew groupoid ring $A{⋊}_{\sigma }G$ is prime if, and only if, A_e is $G_e^e$-prime and $G\prime$ is connected.

Proof.

It follows from Theorem 4.21 and [9, Theorem 13.5]. □

4.3 Groupoid rings

Let R be an s-unital ring and let G be a groupoid. The groupoid ring $R[G]$ consists of elements of the form $\sum _{g\in G}{a}_g{\delta }_g$ where $a_g\in R$ is zero for all but finitely many $g\in G$. For $g,h\in G$ and $a,b\in R$, the multiplication in $R[G]$ is defined by the relation $a{\delta }_g·b{\delta }_h:=\mathit{\text{ab}}{\delta }_{\mathit{\text{gh}}}$, if g, h are composable, and $a{\delta }_g·b{\delta }_h:=0$ otherwise.

Remark 4.24.

Let R be an s-unital ring and let G be a groupoid. Consider the global action $\sigma ={(A_g,{\sigma }_g)}_{g\in G}$ of G on A, defined by letting $A_g:=R$ and ${\sigma }_g:={\text{id}}_R$ for every $g\in G$, and $A:={\oplus }_{e\in G_0}{A}_e={\oplus }_{e\in G_0}R$. Notice that the corresponding skew groupoid ring $A{⋊}_{\sigma }G$ is isomorphic to the groupoid ring $R[G]$.

A subset $X\subseteq G_0$ is said to be R-dense if for every nonzero $a\in R[G]$ there is some $g\in \text{Supp}(a)$ such that $s(g)\in X.$ For each $e\in G_0,$ define ${\mathcal{O}}_e:=\{f\in G_0:\exists g\in G,s(g)=e,r(g)=f\}$.

Proposition 4.25.

Let G be a groupoid and let R be a nonzero s-unital ring. The following statements are equivalent:

1. For every $e\in G_0,$ ${\mathcal{O}}_e$ is R-dense;

2. There is some $e\in G_0$ such that ${\mathcal{O}}_e$ is R-dense;

3. G is connected.

Proof.

(i) $\Rightarrow$ (ii) The proof is immediate.

(ii) $\Rightarrow$ (iii) Suppose that there is some $e\in G_0$ such that ${\mathcal{O}}_e$ is R-dense. Let $f,h\in G_0$ and let $r\in R$ be nonzero. Clearly, $r{\delta }_f,r{\delta }_h\in R[G],$ and hence, by assumption, $f,h\in {\mathcal{O}}_e.$ By the definition of ${\mathcal{O}}_e$, we may find some $g\in G$ such that s(g) = f and $r(g)=h.$

(iii) $\Rightarrow$ (i) Fix $e\in G_0$ and suppose that G is connected. Clearly, ${\mathcal{O}}_e=G_0$ which is R-dense. □

The following theorem generalizes [9, Theorem 12.4].

Theorem 4.26.

Let G be a groupoid and let R be a nonzero s-unital ring. The following statements are equivalent:

1. The groupoid ring $R[G]$ is prime;

2. G is connected and there is some $e\in G_0$ such that the group ring $R[G_e^e]$ is prime;

3. G is connected and, for every $e\in G_0,$ the group ring $R[G_e^e]$ is prime;

4. There is some $e\in G_0$ such that ${\mathcal{O}}_e$ is R-dense and $R[G_e^e]$ is prime;

5. For every $e\in G_0,$ ${\mathcal{O}}_e$ is R-dense and $R[G_e^e]$ is prime;

6. G is connected, R is prime and there is some $e\in G_0$ such that $G_e^e$ has no non-trivial finite normal subgroup;

7. G is connected, R is prime and, for every $e\in G_0,$ $G_e^e$ has no non-trivial finite normal subgroup;

8. R is prime, and there is some $e\in G_0$ such that ${\mathcal{O}}_e$ is R-dense, and $G_e^e$ has no non-trivial finite normal subgroup;

9. R is prime, and for every $e\in G_0,$ ${\mathcal{O}}_e$ is R-dense, and $G_e^e$ has no non-trivial finite normal subgroup.

Proof.

Notice that $G\prime =G.$ The proof follows from Remark 4.24, Theorem 4.21(i), (vi), and (vii), Proposition 4.25 and [9, Theorem 12.4]. □

Remark 4.27.

(a) It is known that, in the case where R is a commutative unital ring, the groupoid ring $R[G]$ is an example of a Steinberg algebra (see [23, Remark 4.10]). Hence, in that special case, the equivalence between (i) and (iv) in Theorem 4.26 can be obtained using Steinberg’s results from [24, Proposition 4.3], [24, Proposition 4.4], and [24, Theorem 4.9].

(b) In the case where R is unital, after suitable translations of the properties involved, it is possible to obtain e.g. the implication (ii) $\Rightarrow$(i) in Theorem 4.26 from [13, Theorem 3.2].

Acknowledgments

The authors are grateful to Patrik Lundström for making them aware of Munn’s result in [14].

Disclosure statement

The authors report that there are no competing interests to declare.

Additional information

Funding

The first named author was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES)