[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.b140895

QUASI-COMPLETENESS AND LOCALIZATIONS OF POLYNOMIAL DOMAINS: A CONJECTURE FROM "OPEN PROBLEMS IN COMMUTATIVE RING THEORY"

Farley, Jonathan David (Department of Mathematics Morgan State University)

Publication Information

Bulletin of the Korean Mathematical Society / v.53, no.6, 2016 , pp. 1613-1615 More about this Journal

Abstract

It is proved that $k[X_1,{\ldots},X_v ]$ localized at the ideal ( $X_1,{\ldots},X_v$ ), where k is a field and $X_1,{\ldots},X_v$ indeterminates, is not weakly quasi-complete for $v{\geq}2$ , thus proving a conjecture of D. D. Anderson and solving a problem from "Open Problems in Commutative Ring Theory" by Cahen, Fontana, Frisch, and Glaz.

Keywords

quasi-completeness; Noetherian ring; commutative ring; polynomial ring; localization; ring of formal power series; completion;

Citations & Related Records

Reference

1	D. D. Anderson, Quasi-complete semilocal rings and modules, In Fontana et al. (eds.), Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, Springer Verlag, New York, 2014.
2	M. Fontana, S. Frisch, and S. Glaz, Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, Springer Verlag, New York, 2014.
3	D. G. Northcott, Ideal Theory, Cambridge University Press, Cambridge, 1953.
4	R. Y. Sharp, Steps in Commutative Algebra, second edition, Cambridge University Press, Cambridge, 2000.
5	P.-J. Cahen, M. Fontana, S. Frisch, and S. Glaz, Open Problems in Commutative Ring Theory, In Fontana et al. (eds.), Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, Springer Verlag, New York, 2014.