Toward Homological Characterization of Semirings by e-Injective Semimodules

TL;DR

This paper introduces e-injective semimodules over semirings, providing characterizations of semirings with all semimodules e-injective, and explores their properties in various algebraic structures.

Contribution

It offers a comprehensive homological framework for semirings via e-injective semimodules, including characterizations of specific classes of semirings.

Findings

Characterization of semirings with all semimodules e-injective

Description of semirings where projective semimodules are e-injective

Characterizations of bounded distributive lattices, subtractive, and simple semirings with e-injective cyclic semimodules

Abstract

In this paper, we introduce and study e-injective semimodules, in particular over additively idempotent semirings. We completely characterize semirings all of whose semimodules are e-injective, describe semirings all of whose projective semimodules are e-injective, and characterize one-sided Noetherian rings in terms of direct sums of e-injective semimodules. Also, we give complete characterizations of bounded distributive lattices, subtractive semirings, and simple semirings, all of whose cyclic (finitely generated) semimodules are e-injective.