Relative injective envelopes and relative projective covers on ring extensions

Abstract

A ring extension is a ring homomorphism preserving identities. In this paper, we give the definitions of relative injective envelopes and relative projective covers of modules on ring extensions, and study their basic properties. In particular, we give their equivalent characterizations in terms of relative essential monomorphisms and relative superfluous epimorphisms, and prove that relative injective envelopes and relative projective covers on ring extensions are unique up to isomorphism whenever they exist. Moreover, for an extension of Artin algebras, we show that every finitely generated module has both a relative injective envelope and a relative projective cover. In addition, we compare relative injective envelopes and relative projective covers on two ring extensions linked by surjective homomorphisms of rings respectively.

KEYWORDS:

• Relative essential monomorphisms
• relative injective envelopes
• relative projective covers
• relative superfluous epimorphisms
• ring extensions

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 18G25
• 16D10

Acknowledgments

The author thanks the referee for helpful comments and suggestions.