Continuous Comodules

TL;DR

This paper extends the concept of clean modules to comodules over coalgebras, showing that continuous comodules are inherently clean, thus broadening the understanding of module and comodule structures in algebra.

Contribution

The paper introduces the notion of clean comodules and proves that all continuous comodules over coalgebras are clean, generalizing existing module theory results.

Findings

Every continuous comodule is a clean comodule.

The concept of clean comodules extends the idea of clean modules to coalgebra contexts.

Provides a new perspective on the structure of comodules in algebra.

Abstract

Let $R$ be a commutative ring with unity and $C$ be an $R$-coalgebra. The ring $R$ is clean if every $ r\in R $ is the sum of a unit and an idempotent element of $R$. An $R$-module $M$ is clean if the endomorphism ring of $M$ over $R$ is clean. Moreover, every continuous module is clean. We modify this idea to the comodule and coalgebra cases. A $C$-comodule $M$ is called a clean comodule if the $C$-comodule endomorphisms of $M$ are clean. We introduced continuous comodules and proved that every continuous comodules is a clean comodule.