Korean J. Math.  Vol 24, No 3 (2016)  pp.537-543
DOI: https://doi.org/10.11568/kjm.2016.24.3.537

When the Nagata ring ${D(X)}$ is a sharp domain

Gyu Whan Chang

Abstract


Let $D$ be an integral domain, $X$ be an indeterminate over $D$, $D[X]$ be the polynomial ring over $D$, and $D(X)$  be the Nagata ring of $D$. Let $[d]$ be the star operation on $D[X]$, which is an extension of the $d$-operation on $D$ as in [5,Theorem 2.3]. In this paper, we show that $D$ is a sharp domain if and only if $D[X]$ is a $[d]$-sharp domain, if and only if $D(X)$ is a sharp domain.

Keywords


star operation, sharp domain, polynomial ring, Nagata ring

Subject classification

13A15, 13F05

Sponsor(s)



Full Text:

PDF

References


Z. Ahmad, T. Dumitrescu, and M. Epure, A Schreier domain type condition, Bull. Math. Soc. Sci. Math. Roumanie 55 (2012), 241–247. (Google Scholar)

Z. Ahmad, T. Dumitrescu, and M. Epure, A Schreier Domain Type Condition II, Algebra Colloquium 22 (2015), 923–934. (Google Scholar)

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

G.W. Chang, Strong Mori domains and the ring D[X]Nv , J. Pure Appl. Algebra 197 (2005), 293–304. (Google Scholar)

G.W. Chang and M. Fontana, Uppers to zero and semistar operations in poly- nomial rings, J. Algebra 318 (2007), 484–493. (Google Scholar)

G.W. Chang and M. Fontana, An overring-theoretic approach to polynomial extensions of star and semistar operations, Comm. Algebra 39 (2011), 1956– 1978. (Google Scholar)

M. Fontana, P. Jara, and E. Santos, Pru ̈fer ∗-multiplication domains and semis- tar operations, J. Algebra Appl. 2 (2003), 1–30. (Google Scholar)

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

J. Hedstrom and E. Houston, Some remarks on star-operations, J. Pure Appl. Algebra 18 (1980), 37–44. (Google Scholar)

B.G. Kang, Pru ̈fer v-multiplication domains and the ring R[X]Nv , J. Algebra 123 (1989), 151–170. (Google Scholar)

A. Mimouni, Note on star operations over polynomial rings, Comm. Algebra 36 (2008), 4249–4256. (Google Scholar)


Refbacks

  • 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