Construction of totally reflexive modules from an exact pair of zero divisors

TL;DR

This paper constructs an infinite family of indecomposable totally reflexive modules over certain local rings with zero divisors, answering a previously posed question and analyzing homomorphism modules.

Contribution

It explicitly constructs infinite non-isomorphic totally reflexive modules from an exact pair of zero divisors under specific conditions, advancing understanding of module theory over such rings.

Findings

Constructed an infinite family of non-isomorphic indecomposable totally reflexive modules.

Provided explicit formulas for homomorphism modules between these modules.

Answered a question posed by Christensen et al. regarding module construction.

Abstract

Let A be a local ring which admits an exact pair x,y of zero divisors as defined by Henriques and Sega. Assuming that this pair is regular and that there exists a regular element on the A-module A/(x,y), we explicitly construct an infinite family of non-isomorphic indecomposable totally reflexive A-modules. In this setting, our construction provides an answer to a question raised by Christensen, Piepmeyer, Striuli, and Takahashi. Furthermore, we compute the module of homomorphisms between any two given modules from the infinite family mentioned above.