n-absorbing ideal factorization in commutative rings

Abstract

In this article, we show that Mori domains, pseudo-valuation domains, and n-absorbing ideals, the three seemingly unrelated notions in commutative ring theory, are interconnected. In particular, we prove that an integral domain R is a Mori locally pseudo-valuation domain if and only if each proper ideal of R is a finite product of 2-absorbing ideals of R. Moreover, every ideal of a Mori locally almost pseudo-valuation domain can be written as a finite product of 3-absorbing ideals. To provide concrete examples of such rings, we study rings of the form $A+\mathit{\text{XB}}[X]$ where A is a subring of a commutative ring B and X is indeterminate, which is of independent interest, and along with several characterization theorems, we prove that in such a ring, each proper ideal is a finite product of n-absorbing ideals for some $n\ge 2$ if and only if $A\subseteq B$ is essentially a finite product of field extensions. A complete description of when an order of a quadratic number field is a locally pseudo valuation domain, a locally almost pseudo valuation domain or a locally conducive domain is given.

KEYWORDS:

• 2-absorbing ideals
• almost pseudo-valuation domains
• Mori domains
• pseudo-valuation domains
• strongly Laskerian rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13A15
• Secondary: 13B30
• 13F05
• 13G05

Acknowledgments

The author wishes to thank the anonymous referee whose careful reading and helpful suggestions, including the current form of Proposition 15, greatly improved the presentation of this manuscript.

Additional information

Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)(No. 2022R1C1C2009021).