Korean J. Math. Vol. 23 No. 4 (2015) pp.631-636
DOI: https://doi.org/10.11568/kjm.2015.23.4.631

A new characterization of Pr\"ufer $v$-multiplication domains

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Gyu Whan Chang


Let $D$ be an integral domain and $w$ be the so-called $w$-operation on $D$. In this note, we introduce the notion of $*(w)$-domains: $D$ is a $*(w)$-domain if $((\cap (x_i))(\cap (y_j)))_w = \cap (x_iy_j)$ for all nonzero elements $x_1, \dots , x_m; y_1, \dots , y_n$ of $D$. We then show that $D$ is a Pr\"ufer $v$-multiplication domain if and only if $D$ is a $*(w)$-domain and $A^{-1}$ is of finite type for all nonzero finitely generated fractional ideals $A$ of $D$.

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