Département de Mathématiques et Informatiques, Faculté des Sciences et Techniques, UCAD, DAKAR, Senegal

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#### 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.