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A characterization of minimal prime ideals

Published online by Cambridge University Press:  18 May 2009

Gary F. Birkenmeier ,
Jin Yong Kim  and
Jae Keol Park
Gary F. Birkenmeier
Affiliation:
Department of Mathematics, University of Southwestern Louisiana, Lafayette, La 70504, USA
Jin Yong Kim
Affiliation:
Department of Mathematics, Kyung Hee University, Suwon 449-701, South Korea
Jae Keol Park
Affiliation:
Department of Mathematics, Busan National University, Busan 609-735, South Korea
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Abstract

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Let P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 40 , Issue 2 , May 1998 , pp. 223 - 236
Copyright
Copyright © Glasgow Mathematical Journal Trust 1998

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