SOME REMARKS ON LOCALLY NOETHERIAN AND LOCALLY ARTINIAN S-ACTS OVER MONOIDS

Abstract

In the category of S-acts, artinian S-acts are introduced as the acts that satisfy the descending chain condition on its congruences. Noetherian S-acts are also the acts that satisfy the ascending chain condition on its congruences. Rees noetherian and Rees artinian case is defined using Rees congruence. This paper is devoted to study the classes of locally artinian, locally artinian, locally Rees artinian and locally Rees artinian S-acts. An S-act is said to be locally artinian if all its finitely generated subacts artinian. The noetherian, Rees noetherian and Rees artinian case is defined similarly. We give some general properties of these classes of S-acts, specially discuss the behaviour of such acts under Rees short exact sequence and coproduct. Finally, we establish some connections between some classes of S-acts such as projectivity and injectivity with the notions related to locally artinian and locally noetherian.