Mamadou Sanghare

Copyright © 1997 Mamadou Sanghare. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let R be a non-commutative associative ring with unity 1≠0, a left R-module is said to satisfy property (I) (resp. (S)) if every injective (resp. surjective) endomorphism of M is an automorphism of M. It is well known that every Artinian (resp. Noetherian) module satisfies property (I) (resp. (S)) and that the converse is not true. A ring R is called a left I-ring (resp. S-ring) if every left R-module with property (I) (resp. (S)) is Artinian (resp. Noetherian). It is known that a subring B of a left I-ring (resp. S-ring) R is not in general a left I-ring (resp. S-ring) even if R is a finitely generated B-module, for example the ring M3(K) of 3×3 matrices over a field K is a left I-ring (resp. S-ring), whereas its subring B={[α00βα0γ0α]/α,β,γ∈K} which is a commutative ring with a non-principal Jacobson radical J=K.[000100000]+K.[000000100] is not an I-ring (resp. S-ring) (see [4], theorem 8). We recall that commutative I-rings (resp S-tings) are characterized as those whose modules are a direct sum of cyclic modules, these tings are exactly commutative, Artinian, principal ideal rings (see [1]). Some classes of non-commutative I-rings and S-tings have been studied in [2] and [3]. A ring R is of finite representation type if it is left and right Artinian and has (up to isomorphism) only a finite number of finitely generated indecomposable left modules. In the case of commutative rings or finite-dimensional algebras over an algebraically closed field, the classes of left I-rings, left S-rings and rings of finite representation type are identical (see [1] and [4]). A ring R is said to be a ring with polynomial identity (P. I-ring) if there exists a polynomial f(X1,X2,…,Xn), n≥2, in the non-commuting indeterminates X1,X2,…,Xn, over the center Z of R such that one of the monomials of f of highest total degree has coefficient 1, and f(a1,a2,…,an)=0 for all a1,a2,…,an in R. Throughout this paper all rings considered are associative rings with unity, and by a module M over a ring R we always understand a unitary left R-module. We use MR to emphasize that M is a unitary right R-module.