Zero-divisors of semigroup modules

TL;DR

This paper investigates the zero-divisors in semigroup modules, establishing conditions under which the zero-divisor structure of the module extension reflects that of the original module, especially for certain types of monoids.

Contribution

It characterizes when semigroup modules have few zero-divisors of a given degree based on properties of the original module and the monoid, extending understanding of zero-divisors in module extensions.

Findings

Zero-divisors of semigroup modules relate to those of the original module.

For certain monoids, the zero-divisor degree of the module extension matches that of the original.

Property (A) is crucial for the equivalence in zero-divisor degrees.

Abstract

Let $M$ be an $R$-module and $S$ a semigroup. Our goal is to discuss zero-divisors of the semigroup module $M[S]$. Particularly we show that if $M$ is an $R$-module and $S$ a commutative, cancellative and torsion-free monoid, then the $R[S]$-module $M[S]$ has few zero-divisors of degree $n$ if and only if the $R$-module $M$ has few zero-divisors of degree $n$ and Property (A).