Semicorings and Semicomodules

TL;DR

This paper introduces semicorings over semirings and explores their semicomodule categories, extending classical coring theory to a more general, non-Abelian setting with new categorical and homological methods.

Contribution

It generalizes coring and comodule theory from rings to semirings, addressing the challenges posed by the non-Abelian base category with novel methods.

Findings

Development of the theory of semicorings over semirings.

Introduction of a new notion of exact sequences for semimodules.

Extension of classical coring results to a broader algebraic context.

Abstract

In this paper, we introduce and investigate \emph{semicorings} over associative semirings and their categories of \emph{semicomodules.} Our results generalize old and recent results on corings over rings and their categories of comodules. The generalization is \emph{not} straightforward and even subtle at some places due to the nature of the base category of commutative monoids which is neither Abelian (not even additive) nor homological, and has no non-zero injective objects. To overcome these and other difficulties, a combination of methods and techniques from categorical, homological and universal algebra is used including a new notion of exact sequences of semimodules over semirings.