Annales Mathematiques Africaines


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On Commutative Goldie I-Rings and Commutative Goldie S-Rings




Abdoulaye MBAYE


Mamadou SANGHARÉ


Sidy Demba TOURÉ


Département de Mathématique et Informatique
Faculté des Sciences et Techniques
Université Cheikh Anta Diop de Dakar, SÉNÉGAL



Mathematics Subject Classifications: 16G10, 16D50.
Key words : finite Goldie dimensional, Goldie I-rings, Goldie S-rings, property (I), property (S), semiartinian.

Abstract:


Let R be a commutative ring. An unital R-module M is said to have property (I) (resp. property(S)) if every injective (resp. surjective) endomorphism of M is an automorphism. An unital R-module M is called finite Goldie dimensional if M contains no infinite direct sum of nonzero submodules. The commutative ring R is called commutative Goldie I-ring (resp. S-ring) if every R-module with property (I) (resp. property (S)) is finite Goldie dimensional. In this note we show that for a commutative semiartinian ring the following conditions are equivalent:

(i)  R is an artinian principal ideal ring;

(ii)  R is a Goldie I-ring;

(iii)  R is a Goldie S-ring.

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