Module-Theoretic Characterizations of Prufer $v$-Multiplication Domains

We present unified $w$-theoretic characterizations of Prüfer $v$-multiplication domains (P$v$MDs). A module-theoretic perspective shows that torsion submodules are $w$-pure, and for $(w$-)$$finitely generated modules $M$, the canonical sequence $0\to T(M)\to M\to M\mathrm{/}T(M)\to 0$ $w$-splits, resolving an open question of Geroldinger–Kim–Loper. In a $w$-version of Hattori-Davis theory, these conditions are equivalent to $To{r}_2^R(M,N)$ being $GV$-torsion for all $R$-modules $M,N$, equivalently $w$-w.gl.dim$(R)\le 1$, or $To{r}_1^R(X,A)$ being $GV$-torsion for all $X$ and torsion-free $A$, or the Davis map $A{\otimes }_{R}B\to \mathcal{T}{\otimes }_K\mathcal{S}$ having $GV$-torsion kernel. From an overring viewpoint, $R$ is a P$v$MD if and only if for every $R\subseteq T\subseteq K$ and every $w$-maximal ideal $m$, the localization $R_m\to T_{\text{m}}$ is a flat epimorphism, so that each overring is $w$-flat and the inclusion is $w$-epimorphic. Finally, $R$ is a P$v$MD if and only if every pure $w$-injective divisible $R$-module is injective.