We present unified w-theoretic characterizations of Prüfer v-multiplication domains (PvMDs). A module-theoretic perspective shows that torsion submodules are w-pure, and for (w-) finitely generated modules M, the canonical sequence <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0\rightarrow T(M)\rightarrow M\rightarrow M/T(M)\rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo stretchy="false">→</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">/</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> w-splits, resolving an open question of Geroldinger–Loper–Kim. In a w-version of Hattori–Davis theory, these conditions are equivalent to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textrm{Tor}^R_2(M,N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mtext>Tor</mtext> <mn>2</mn> <mi>R</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> being <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{GV}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>GV</mtext> </math></EquationSource> </InlineEquation>-torsion for all R-modules M, N, equivalently w-w.gl.dim<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(R)\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, or <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{Tor}^R_1(X,A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mtext>Tor</mtext> <mn>1</mn> <mi>R</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> being <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textrm{GV}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>GV</mtext> </math></EquationSource> </InlineEquation>-torsion for all X and torsion-free A, or the Davis map <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$A\otimes _R B \rightarrow \mathcal {T}\otimes _K \mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <msub> <mo>⊗</mo> <mi>R</mi> </msub> <mi>B</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">T</mi> <msub> <mo>⊗</mo> <mi>K</mi> </msub> <mi mathvariant="script">S</mi> </mrow> </math></EquationSource> </InlineEquation> having <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textrm{GV}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>GV</mtext> </math></EquationSource> </InlineEquation>-torsion kernel. From an overring viewpoint, R is a PvMD if and only if for every <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$R\subseteq T\subseteq K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>⊆</mo> <mi>T</mi> <mo>⊆</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> and every w-maximal ideal <InlineEquation ID="IEq10"> <EquationSource Format="TEX">${\mathfrak {m}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">m</mi> </math></EquationSource> </InlineEquation>, the localization <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$R_{{\mathfrak {m}}}\rightarrow T_{{\mathfrak {m}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi mathvariant="fraktur">m</mi> </msub> <mo stretchy="false">→</mo> <msub> <mi>T</mi> <mi mathvariant="fraktur">m</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is a flat epimorphism, so that each overring is w-flat and the inclusion is w-epimorphic. Finally, R is a PvMD if and only if every pure w-injective divisible R-module is injective.

Module-Theoretic Characterizations of Prüfer v-Multiplication Domains

期刊论文
发表日期：

2026年1月30日

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Xiaolei Zhang,
Hwankoo Kim

摘要

We present unified w-theoretic characterizations of Prüfer v-multiplication domains (PvMDs). A module-theoretic perspective shows that torsion submodules are w-pure, and for (w-) finitely generated modules M, the canonical sequence $0\rightarrow T(M)\rightarrow M\rightarrow M/T(M)\rightarrow 0$ $0 \to T (M) \to M \to M / T (M) \to 0$ w-splits, resolving an open question of Geroldinger–Loper–Kim. In a w-version of Hattori–Davis theory, these conditions are equivalent to $\textrm{Tor}^R_2(M,N)$ ${Tor}_{2}^{R} (M, N)$ being $\textrm{GV}$ $GV$ -torsion for all R-modules M, N, equivalently w-w.gl.dim $(R)\le 1$ $(R) \leq 1$ , or $\textrm{Tor}^R_1(X,A)$ ${Tor}_{1}^{R} (X, A)$ being $\textrm{GV}$ $GV$ -torsion for all X and torsion-free A, or the Davis map $A\otimes _R B \rightarrow \mathcal {T}\otimes _K \mathcal {S}$ $A \otimes_{R} B \to T \otimes_{K} S$ having $\textrm{GV}$ $GV$ -torsion kernel. From an overring viewpoint, R is a PvMD if and only if for every $R\subseteq T\subseteq K$ $R \subseteq T \subseteq K$ and every w-maximal ideal ${\mathfrak {m}}$ $m$ , the localization $R_{{\mathfrak {m}}}\rightarrow T_{{\mathfrak {m}}}$ $R_{m} \to T_{m}$ is a flat epimorphism, so that each overring is w-flat and the inclusion is w-epimorphic. Finally, R is a PvMD if and only if every pure w-injective divisible R-module is injective.