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

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 $D$.


Pr\"ufer $v$-multiplication domain; $(t,v)$-Dedekind domain; $*(w)$-domain

Subject classification

13A15, 13F05


Full Text:



D.D. Anderson, D.F. Anderson, M. Fontana, and M. Zafrullah, On v-domains and star operations, Comm. Algebra 37 (2009), 3018–3043. (Google Scholar)

D.D. Anderson and S.J. Cook, Two star-operations and their induced lattices, Comm. Algebra 28 (2000), 2461–2475. (Google Scholar)

R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972. (Google Scholar)

Q. Li, (t, v)-Dedekind domains and the ring R[X]Nv , Results in Math. 59 (2011), 91–106. (Google Scholar)

M. Zafrullah, On generalized Dedekind domains, Mathematika 33 (1986), 285–295. (Google Scholar)

M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra 15 (1987), 1895–1920. (Google Scholar)

M. Zafrullah, Ascending chain condition and star operations, Comm. Algebra 17 (1989), 1523–1533. (Google Scholar)


  • There are currently no refbacks.

ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr