# On zero-divisors of semimodules and semialgebras

## TL;DR

This paper explores zero-divisors in semimodules and semialgebras, establishing properties like McCoy's property, and introduces concepts such as Auslander semimodules and strong Krull primes, advancing the algebraic theory of semirings.

## Contribution

It proves McCoy's property for polynomial zero-divisors in semirings, characterizes zero-divisors in monoid semimodules, and introduces new classes like Auslander and strong Krull primes.

## Key findings

- McCoy's property holds for polynomial zero-divisors in semirings
- Zero-divisors in monoid semimodules relate to those in base semimodules
- Introduction of strong Krull primes for semirings

## Abstract

In Section 1 of the paper, we prove McCoy's property for the zero-divisors of polynomials in semirings. We also investigate zero-divisors of semimodules and prove that under suitable conditions, the monoid semimodule $M[G]$ has very few zero-divisors if and only if the $S$-semimodule $M$ does so. The concept of Auslander semimodules are introduced in this section as well. In Section 2, we introduce Ohm-Rush and McCoy semialgebras and prove some interesting results for prime ideals of monoid semirings. In Section 3, we investigate the set of zero-divisors of McCoy semialgebras. We also introduce strong Krull primes for semirings and investigate their extension in semialgebras.