Chain conditions on (Rees) congruences of S-acts

TL;DR

This paper explores chain conditions on Rees congruences of S-acts, introducing Rees artinian and noetherian acts, and examines their properties and inheritance in Rees short exact sequences, with applications to monoids.

Contribution

It introduces Rees artinian and noetherian S-acts, studies their properties, inheritance, and applies these concepts to monoids, extending the theory of chain conditions.

Findings

Rees artinian S-acts satisfy the descending chain condition on Rees congruences.

All factor acts of Rees artinian S-acts are finitely Rees cogenerated.

Every right Rees artinian, commutative monoid is right Rees noetherian.

Abstract

In this paper, we introduce (Rees) artinian S-acts as those acts that satisfy the descending chain condition on their (Rees) congruences. Then, we show that artinian S-acts are those acts whose all factor acts are finitely (Rees) cogenerated. In addition, we continue the study of (Rees) noetherian S-acts, namely, those acts whose (Rees) congruences are finitely generated. Moreover, as a useful tool for the investigation of chain conditions, we prove that the properties of being (Rees) noetherian and artinian are inherited in Rees short exact sequences. Next, we specifically consider the chain conditions on monoids. Finally, we prove that every right Rees artinian, commutative monoid is right Rees noetherian.