Contents > Past issues > Volume 1
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.