# Pushouts and e-Projective Semimodules

## TL;DR

This paper explores the concept of e-projective semimodules over semirings, examining their properties and the role of pushouts, with a focus on new notions of exact sequences and constructive proofs.

## Contribution

It introduces the notion of e-projective semimodules using new exact sequence concepts and provides a constructive proof of pushout existence in semimodule categories.

## Key findings

- Characterization of e-projective semimodules
- Constructive proof of pushout existence
- Insights into exact sequences of semimodules

## Abstract

Projective modules play an important role in the study of the category of modules over rings and in the characterization of various classes of rings. Several characterizations of projective objects which are equivalent for modules over rings are not necessarily equivalent for semimodules over an arbitrary semiring. We study several of these notions, in particular the e-projective semimodules introduced by the first author using his new notion of exact sequences of semimodules. As pushouts of semimodules play an important role in some of our proofs, we investigate them and give a constructive proof of their existence in a way that proved be very helpful.