The purpose of this paper is to study some results of homological algebra in the category A-Alg (resp. Alg-A) of left (resp. right) A-algebra in the noncommutative case. In this paper A is a subring of B. So the main results of this paper are, if B is a noetherian duo-ring, S a central saturated multiplicatively closed subset of A, S R the set of regular elements of S, " a finitely presented right A-algebra and a (B-A)-bialgebra, then ",
Let R be a commutative ring. It is known that any injective endomorphism of finitely generated R-module is an isomorphism if and only if every prime ideal of R is maximal. This result makes possible a characterization of rings on which all finitely generated modules are co-hopfian. The motivation of this paper comes from trying to extend these results to mono-correct modules. In doing so, we show that any finitely generated R-module is mono-correct if and only if every prime ideal of R is maximal and we obtain a characterization of commutative rings on which all finitely generated module are mono-correct. Such rings are exactly commutative strongly Π-regular rings. So we have a new characterization of commutative strongly Π-regular rings.
The main result of this paper is the following: Let B be a ring, A a subring of B, M a left finite type A-module and S a saturated multiplicative subset of A satisfying the left conditions of Ore, S −1 A the ring of the A fractions in S, then Ext n S −1 B (S −1 M, −) and T or S −1 A n (S −1 M, −) are adjoint functors.
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