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