Multiplication Semimodules

TL;DR

This paper studies multiplication semimodules over semirings, exploring their properties, generalizing known module results, and characterizing finitely generated cancellative cases under specific conditions.

Contribution

It extends the theory of multiplication modules to semimodules, providing new characterizations and properties, especially for finitely generated and cancellative cases.

Findings

Every multiplicatively cancellative multiplication semimodule is finitely generated and projective.

Characterization of finitely generated cancellative multiplication semimodules over yoked semirings.

Generalization of multiplication module results to the semimodule setting.

Abstract

Let $S$ be a semiring. An $S$-semimodule $M$ is called a multiplication semimodule if for each subsemimodule $N$ of $M$ there exists an ideal $I$ of $S$ such that $N=IM$. In this paper we investigate some properties of multiplication semimodules and generalize some results on multiplication modules to semimodules. We show that every multiplicatively cancellative multiplication semimodule is finitely generated and projective. Moreover, we characterize finitely generated cancellative multiplication $S$-semimodules when $S$ is a yoked semiring such that every maximal ideal of $S$ is subtractive.