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A variant theory for the Gorenstein flat dimension

Samir Bouchiba (2015)

Colloquium Mathematicae

This paper discusses a variant theory for the Gorenstein flat dimension. Actually, since it is not yet known whether the category (R) of Gorenstein flat modules over a ring R is projectively resolving or not, it appears legitimate to seek alternate ways of measuring the Gorenstein flat dimension of modules which coincide with the usual one in the case where (R) is projectively resolving, on the one hand, and present nice behavior for an arbitrary ring R, on the other. In this paper, we introduce...

About G-rings

Najib Mahdou (2017)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we are concerned with G-rings. We generalize the Kaplansky’s theorem to rings with zero-divisors. Also, we assert that if R T is a ring extension such that m T R for some regular element m of T , then T is a G-ring if and only if so is R . Also, we examine the transfer of the G-ring property to trivial ring extensions. Finally, we conclude the paper with illustrative examples discussing the utility and limits of our results.

Contracting endomorphisms and dualizing complexes

Saeed Nasseh, Sean Sather-Wagstaff (2015)

Czechoslovak Mathematical Journal

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring R . Our focus is on homological properties of contracting endomorphisms of R , e.g., the Frobenius endomorphism when R contains a field of positive characteristic. For instance, in this case, when R is F -finite and C is a semidualizing R -complex, we prove that the following conditions are equivalent: (i) C is a dualizing R -complex; (ii) C 𝐑 Hom R ( n R , C ) for some n > 0 ; (iii) G C -dim n R < and C is derived 𝐑 Hom R ( n R , C ) -reflexive...

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