# Injective Semimodules - Revisited

## TL;DR

This paper explores the properties and relationships of various types of injective semimodules, introducing new notions and partial solutions to embedding problems, thereby advancing the understanding of semimodule theory.

## Contribution

It introduces a new notion of exact sequences for semimodules and clarifies relationships among different injective semimodule concepts, including partial solutions to the embedding problem.

## Key findings

- Clarified relationships between injective, e-injective, and i-injective semimodules
- Provided examples and counterexamples illustrating these relationships
- Presented partial results on the embedding problem for semimodules

## Abstract

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no non-zero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so called e-injective semimodules introduced by the first author using a new notion of exact sequences of semimodules. We clarify the relationships between the injective semimodules, the e-injective semimodule, and the i-injective semimodules through several implications, examples and counter examples. Moreover, we provide partial results for the so called Embedding Problem (of semimodules in injective semimodules).