Coproduct Cancellation on \textbf{Act}-$S$

TL;DR

This paper explores the cancellation property in the theory of acts over monoids, establishing conditions under which acts are cancellable and showing the equivalence of cancellation and internal cancellation.

Contribution

It introduces the cancellation problem for acts over monoids and characterizes when acts are cancellable based on their decomposition into indecomposable subacts.

Findings

Cancellable acts correspond to finite equivalence classes of isomorphic indecomposable subacts.

Every act is cancellable if and only if it is internally cancellable.

The concepts of cancellation and internal cancellation coincide for acts over monoids.

Abstract

The themes of cancellation, internal cancellation, substitution have led to a lot of interesting research in the theory of modules over commutative and noncommutative rings. In this paper, we introduce and study cancellation problem in the theory of acts over monoids. We show that if $A$ is an $S$-act and $A={\dot\bigcup_{i\in I}}A_i$ is the unique decomposition of $A$ into indecomposable subacts $A_i, i\in I$ such that the set $P=\{{\rm Card} [i] \mid i\in I\}$ is finite, then $A$ is cancellable if and only if the equivalence class $[i]=\{j\in I \mid A_i\cong A_j\}$ is finite, for every $i\in I$. Likewise, we prove that every $S$-act is cancellable if and only if it is internally cancellable. Thus, the concepts cancellation and internal cancellation coincide here.