• 대한수학회지
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• 제60권3호
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• pp.683-694
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• 2023

Let a be an ideal of 𝔞 commutative noetherian ring R. We give some descriptions of the 𝔞-depth of 𝔞-relative Cohen-Macaulay modules by cohomological dimensions, and study how relative Cohen-Macaulayness behaves under flat extensions. As applications, the perseverance of relative Cohen-Macaulayness in a polynomial ring, formal power series ring and completion are given.

• 호남수학학술지
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• 제20권1호
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• pp.1-14
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• 1998

In this short note we study the torsion theories over a commutative ring R and discuss a relative dimension related to such theories for R-modules. Let ${\sigma}$ be a torsion functor and (T, F) be its corresponding partition of Spec(R). The concept of ${\sigma}$-Cohen Macaulay (abbr. ${\sigma}$-CM) module is defined and some of the main points concerning the usual Cohen-Macaulay modules are extended. In particular it is shown that if M is a non-zero ${\sigma}$-CM module over R and S is a multiplicatively closed subset of R such that, for all minimal element of T, $S{\cap}p={\emptyset}$, then $S^{-1}M$ is a $S^{-1}{\sigma}$-CM module over $S^{-1}$R, where $S^{-1}{\sigma}$ is the direct image of ${\sigma}$ under the natural ring homomorphism $R{\longrightarrow}S^{-1}R$.

• 대한수학회보
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• 제52권1호
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• pp.137-147
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• 2015

We derive in the paper the tensor product functor -${\otimes}_R$- by using proper $\mathcal{GP}_C$-resolutions, where C is a semidualizing module. After giving several cases in which different relative homologies agree, we use the Pontryagin duals of $\mathcal{G}_C$-projective modules to establish a balance result for such relative homology over a Cohen-Macaulay ring with a dualizing module D.