Journal of the
Korean Mathematical Society JKMS

Let $R$ be a commutative ring with identity. An $R$-module $M$ is said to be $w$-projective if $\Ext_R^1(M,N)$ is GV-torsion for any torsion-free $w$-module $N$. In this paper, we define a ring $R$ to be $w$-semi-hereditary if every finite type ideal of $R$ is $w$-projective. To characterize $w$-semi-hereditary rings, we introduce the concept of $w$-injective modules and study some basic properties of $w$-injective modules. Using these concepts, we show that $R$ is $w$-semi-hereditary if and only if the total quotient ring $T(R)$ of $R$ is a von Neumann regular ring and $R_\fkm$ is a valuation domain for any maximal $w$-ideal $\fkm$ of $R$. It is also shown that a connected ring $R$ is $w$-semi-hereditary if and only if $R$ is a Pr\"ufer $v$-multiplication domain.

Keywords: $w$-projective module, $w$-flat module, $w$-injective module, finite type, $w$-semi-hereditary ring

MSC numbers: 13A15, 13C11, 13D07