JP Journal of Algebra, Number Theory and Applications
Volume 7, Issue 2, Pages 245 - 250
(April 2007)
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SUBRINGS OF FGI-RINGS
Mamadou Barry (S鮩gal), Mamadou Sanghare (S鮩gal ) and Sidy Demba Toure (S鮩gal)
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Abstract: Let R be a noncommutative associative ring with unity A left R-module M is said to have 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 the converse is not true. A ring R is called left I-ring (resp. S-ring) if every left R-module with property (I) (resp. (S)) is Artinian (resp. Noetherian). It is well known that on a commutative ring (resp. commutative ring whose prime ideals are maximal) every finitely generated module satisfies property (I) (resp. (S)) and the converse is not true (see [1]). A ring R is called left (right) FGI-ring if every left (right) R-module with property (I) is finitely generated. R is called FGI-ring if it is both a left and right FGI-ring. If R is either commutative or a duo ring, then the classes of S-rings, I-rings, FGS-rings and FGI-rings are exactly the class of Artinian principal ideal rings (see [2], [5] and [9]). Let R be an integral domain and K be its classical uotient field. If then K is a FGI-ring but R is not a FGI-ring. It is known that a subring B of a left FGI-ring is not in general a left FGI-ring even if R is a finitely generated B-module, for example the ring of matrices over a field K is a left FGI-ring whereas its subring which is a commutative ring with a non principal Jacobson radical is not a FGI-ring (see [6, Theorem 8]). A ring R is said to be a ring with polynomial identity (P.I-ring) if there exists a polynomial in the non commuting indeterminates over the center Z of R such that one of the monomials of f of the highest total degree has coefficient 1, and for all 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 to emphasize that M is a unitary right R-module. The main result of this note is the following Theorem:
Let R be a left Artinian FGI-ring and B be a subring of R contained in the center Z of R. Suppose that R is a finitely generated flat B-module. Then B is a FGI-ring. |
Keywords and phrases: left FGI-ring, ring with polynomial identity, finitely generated module. |
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