• 대한수학회지
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• 제50권1호
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• pp.219-229
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• 2013

Let R be a commutative ring with $1{\neq}0$ and let $\mathcal{H}$ = {R|R is a commutative ring and Nil(R) is a divided prime ideal}. If $R{\in}\mathcal{H}$, then R is called a ${\phi}$-ring. In this paper, we introduce the concepts of ${\phi}$-torsion modules, ${\phi}$-flat modules, and ${\phi}$-von Neumann regular rings.

• 대한수학회보
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• 제58권4호
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• pp.1039-1052
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• 2021

In this paper, we introduce and study the class of 𝜙-w-flat modules which are generalizations of both 𝜙-flat modules and w-flat modules. The 𝜙-w-weak global dimension 𝜙-w-w.gl.dim(R) of a commutative ring R is also introduced and studied. We show that, for a 𝜙-ring R, 𝜙-w-w.gl.dim(R) = 0 if and only if w-dim(R) = 0 if and only if R is a 𝜙-von Neumann ring. It is also proved that, for a strongly 𝜙-ring R, 𝜙-w-w.gl.dim(R) ≤ 1 if and only if each nonnil ideal of R is 𝜙-w-flat, if and only if R is a 𝜙-PvMR, if and only if R is a PvMR.