Subrings of I-rings and S-rings

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 M 3 ( K ) of 3 × 3 matrices over a field K is a left I-ring (resp. S-ring), whereas its subring B = { [ α 0 0 β α 0 γ 0 α ] / α , β , γ ∈ K } which is a commutative ring with a non-principal Jacobson radical J = K . [ 0 0 0 1 0 0 0 0 0 ] + K . [ 0 0 0 0 0 0 1 0 0 ] 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 ( X 1 , X 2 , … , X n ) , n ≥ 2 , in the non-commuting indeterminates X 1 , X 2 , … , X n , over the center Z of R such that one of the monomials of f of highest total degree has coefficient 1 , and f ( a 1 , a 2 , … , a n ) = 0 for all a 1 , a 2 , … , a n 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 M R to emphasize that M is a unitary right R -module.