Commutative Ring Theory

17 A Characterization of Semi-Artinian Rings

ABDELOUAHAB IDELHADJ Département de Mathématiques, Faculté des Sciences de Tétouan, Tétouan, Morocco

AMINE KAIDI Département de Mathématiques, Faculté des Sciences de Rabat, Rabat, Morocco

Introduction

Let A be a commutative ring and M an A-module. The socle of M is the sum of simple submodules of M, the socle of A is the socle of the A-module A. The socle of an A-module is an invariant of its isomorphy class but does not determine it. That is to say, if M and N are two isomorphic A-modules then soc M and soc N are isomorphic, but the inverse is not true in general. So in the category of groups considered as Z-modules we have :

$$\mathrm{Soc}\left({\text{Q}}_{\text{p}}\text{/Z}\right)\approx \mathrm{Soc}\left({\text{Z/p}}^{\text{n}}\text{Z}\right)\approx (\text{Z/pZ)}\text{ and }\mathrm{Soc}\text{Q}\approx \mathrm{Soc}Z$$

where Q is the set of rational ...