# Totally reflexive modules over connected sums with m^3=0

## TL;DR

This paper provides a criterion for identifying rings with nilpotent maximal ideal cube, formed as connected sums, that admit non-trivial totally acyclic modules, advancing understanding of their homological properties.

## Contribution

It introduces a new criterion for rings with =0 formed as connected sums to possess non-trivial totally acyclic modules, expanding the classification of such rings.

## Key findings

- Rings with =0 can have non-trivial totally acyclic modules under specific conditions.
- Connected sums of rings can be characterized for the existence of these modules.
- The paper establishes a link between ring structure and homological properties.

## Abstract

We give a criterion for rings with $\m^3=0$ which are obtained as connected sums of two other rings to have non-trivial totally acyclic modules.