SUBRINGS OF I-RINGS AND S-RINGS

MAMADOU SANGHARE

《International Journal of Mathematics and Mathematical Sciences》 1997卷 (1997/12) Pp. 825-827

https://doi.org/10.1155/S0161171297001130

摘要

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 x 3 matrices over a field K is a left I-ring (resp S-ring), whereas its subring (The equation is abbreviated) which is a commutative ring with a non-principal Jacobson radical (The equation is abbreviated) 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 rings are exactly commutative, Artinian, principal ideal rings (see [1]). Some classes of non-commutative I-rings and Stings have been studied in [2] and [3]. A ring R is of finite representation type if it is left and fight Artinian and has (up to isomorphism) only a finite number of finitely generated indecomposable left modules. In the case of commutative tings or finite-dimensional algebras over an algebraically closed field, the classes of left I-tings, left S-tings and tings 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 fight R-module.

關鍵字

Left I-ring ； left S-ring ； ring with polynomial idemity ； ring of finite representation type