Subrings of I-rings and S-rings

Abstract

LetRbe a non-commutative associative ring with unity1≠0, a leftR-module is said to satisfy property (I) (resp. (S)) if every injective (resp. surjective) endomorphism ofMis an automorphism ofM. It is well known that every Artinian (resp. Noetherian) module satisfies property (I) (resp. (S)) and that the converse is not true. A ringRis called a left I-ring (resp. S-ring) if every leftR-module with property (I) (resp. (S)) is Artinian (resp. Noetherian). It is known that a subringBof a left I-ring (resp. S-ring)Ris not in general a left I-ring (resp. S-ring) even ifRis a finitely generatedB-module, for example the ringM3(K)of3×3matrices over a fieldKis a left I-ring (resp. S-ring), whereas its subringB={[α00βα0γ0α]/α,β,γ∈K}which is a commutative ring with a non-principal Jacobson radicalJ=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 ringRis 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 ringRis said to be a ring with polynomial identity (P. I-ring) if there exists a polynomialf(X1,X2,…,Xn),n≥2, in the non-commuting indeterminatesX1,X2,…,Xn, over the centerZofRsuch that one of the monomials offof highest total degree has coefficient1, andf(a1,a2,…,an)=0for alla1,a2,…,aninR. Throughout this paper all rings considered are associative rings with unity, and by a moduleMover a ringRwe always understand a unitary leftR-module. We useMRto emphasize thatMis a unitary rightR-module.