• Honam Mathematical Journal
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• v.45 no.1
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• pp.146-159
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• 2023

In this paper, we define ⊕_δ-co-coatomically supplemented and co-coatomically δ-semiperfect modules as a strongly notion of ⊕-co-coatomically supplemented and co-coatomically semiperfect modules with the help of Zhou's radical. We say that a module A is ⊕_δ-co-coatomically supplemented if each co-coatomic submodule of A has a δ-supplement in A which is a direct summand of A. And a module A is co-coatomically δ-semiperfect if each coatomic factor module of A has a projective δ-cover. Also we define co-coatomically amply δ-supplemented modules and we examined the basic properties of these modules. Furthermore, we give a ring characterization for our modules. In particular, a ring R is δ-semiperfect if and only if each free R-module is co-coatomically δ-semiperfect.

• Journal of the Korean Mathematical Society
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• v.54 no.6
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• pp.1829-1839
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• 2017

In this paper, we show some results on the vanishing and the finiteness of local cohomology modules with respect to a pair of ideals. We also prove that Supp($H^{dim\;M-1}_{I,J}(M)/JH^{dim\;M-1}_{I,J}(M)$) is a finite set.