Subrings of I-rings and S-rings - Mamadou Sanghare (International Journal of Mathematics and Mathematical Sciences)
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.