Chain conditions on (Rees) congruences of S-acts

Abstract

In this paper, we introduce (Rees) artinian [Formula: see text]-acts as those acts that satisfy the descending chain condition on their (Rees) congruences. Then, we show that (Rees) artinian [Formula: see text]-acts are those acts whose factor acts are all finitely (Rees) cogenerated. In addition, we continue the study of (Rees) noetherian [Formula: see text]-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.