Journal of Commutative Algebra

Star operations on Prüfer v -multiplication domains

Gyu Whan Chang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $D$ be an integrally closed domain, $S(D)$ the set of star operations on $D$, $w$ the $w$-operation, and $S_w(D) = \{* \in S(D) \mid w \leq *\}$. Let $X$ be an indeterminate over $D$ and $N_v = \{f \in D[X] \mid c(f)_v = D\}$. In this paper, we show that, if $D$ is a Pr\"ufer $v$-multiplication domain (P$v$MD), then $|S_w(D)| = |S_w(D[X])| = |S(D[X]_{N_v})|$. We prove that $D$ is a P$v$MD if and only if $|\{* \in S_w(D) \mid *$ is of finite type$\}|\lt \infty$. We then use these results to give a complete characterization of integrally closed domains $D$ with $|S_w(D)| \lt \infty$.

Article information

J. Commut. Algebra, Volume 7, Number 4 (2015), 523-543.

First available in Project Euclid: 19 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A15: Ideals; multiplicative ideal theory 13G05: Integral domains

Star operation $w$-operation P$v$MD $D[X]_{N_v}$ integrally closed domain


Chang, Gyu Whan. Star operations on Prüfer v -multiplication domains. J. Commut. Algebra 7 (2015), no. 4, 523--543. doi:10.1216/JCA-2015-7-4-523.

Export citation


  • D.D. Anderson and S.J. Cook, Two star-operations and their induced lattices, Comm. Alg. 28 (2000), 2461–2475.
  • G.W. Chang, Strong Mori domains and the ring $D[X]_{N_v}$, J. Pure Appl. Alg. 197 (2005), 293–304.
  • ––––, $*$-Noetherian domains and the ring $D[X]_{N_*}$, J. Alg. 297 (2006), 216–233.
  • ––––, Prüfer $*$-multiplication domains, Nagata rings, and Kronecker function rings, J. Alg. 319 (2008), 309–319.
  • G.W. Chang, M. Fontana and M.H. Park, Polynomial extensions of semistar operations, J. Alg. 390 (2013), 250–263.
  • G.W. Chang, E. Houston and M.H. Park, Star operations on strong Mori domains, Houston J. Math.. to appear.
  • S. El Baghdadi and S. Gabelli, $w$-Divisorial domains, J. Alg. 285 (2005), 335–355.
  • M. Fontana, P. Jara and E. Santos, Prüfer $*$-multiplication domains and semistar operations, J. Alg. Appl. 2 (2003), 21–50.
  • M. Fontana and K.A. Loper, Nagata rings, Kronecker function rings and related semistar operations, Comm. Alg. 31 (2003), 4775–4805.
  • S. Gabelli, E. Houston and G. Picozza, $w$-Divisoriality in polynomial rings, Comm. Alg. 37 (2009), 1117–1127.
  • R. Gilmer, Multiplicative ideal theory, Queen's Papers Pure Appl. Math. 90, Queen's University, Kingston, Ontario, 1992.
  • J. Hedstrom and E. Houston, Some remarks on star-operations, J. Pure Appl. Alg. 18 (1980), 37–44.
  • W. Heinzer, Integral domains in which each nonzero ideal is divisorial, Mathematika 15 (1968), 164–170.
  • W. Heinzer, J.A. Huckaba and I. Papick, $m$-canonical ideals in integral domains, Comm. Alg. 26 (1998), 3021–3043.
  • E. Houston, S. Malik and J. Mott, Characterizations of $*$-multiplication domains, Canad. Math. Bull. 27 (1984), 48–52.
  • E. Houston, A. Mimouni and M.H. Park, Integral domains which admit at most two star operations, Comm. Alg. 39 (2011), 1907–1921.
  • ––––, Integrally closed domains with only finitely many star operations, Comm. Alg. 42 (2014), 5264–5286.
  • E. Houston and M. Zafrullah, Integral domains in which each $t$-ideal is divisorial, Michigan Math. J. 35 (1988), 291–300.
  • ––––, On $t$-invertibility II, Comm. Alg. 17 (1989), 1955–1969.
  • B.G. Kang, Prüfer $v$-multiplication domains and the ring $R[X]_{N_v}$, J. Alg. 123 (1989), 151–170.
  • ––––, On the converse of a well-known fact about Krull domains, J. Alg. 124 (1989), 284–299.
  • A. Mimouni, Integral domains in which each ideal is a $w$-ideal, Comm. Alg. 33 (2005), 1345–1355.
  • G. Picozza and F. Tartarone, When the semistar operation $\tilde{\star}$ is the identity, Comm. Alg. 36 (2008), 1954–1975.