### A calculation of injective dimension over valuation domains

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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...

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 $R\subseteq T$ is a ring extension such that $mT\subseteq R$ 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.

Let $R$ be a commutative ring and $\mathcal{C}$ a semidualizing $R$-module. We investigate the relations between $\mathcal{C}$-flat modules and $\mathcal{C}$-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.

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) $C\sim \mathbf{R}{\mathrm{Hom}}_{R}{(}^{n}R,C)$ for some $n>0$; (iii) ${\mathrm{G}}_{C}{\text{-dim}}^{n}R<\infty $ and $C$ is derived $\mathbf{R}{\mathrm{Hom}}_{R}{(}^{n}R,C)$-reflexive...

Let $R$ be a commutative Noetherian ring and $M$ be a finitely generated $R$-module. The main result of this paper is to characterize modules whose first nonzero Fitting ideal is a product of maximal ideals of $R$, in some cases.