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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

References

1.Andrunakievic, V. A. and Rjabuhin, Ju. M., Rings without nilpotent elements and completely simple ideals, Dokl. Akad. Nauk. SSSR. 180 (1968), 9–11 (Translation, Soviet Math. Dokl. 9 (1968), 565–568).Google Scholar
2.Baer, R., Radical ideals, Amer. J. Math. 65 (1943), 537–568.CrossRefGoogle Scholar
3.Birkenmeier, G. F. and Heatherly, H. E., Permutation identity rings and the medial radical, in Non-Commutative Ring Theory, Proceedings of Conference at Athens, Ohio 1989, Lecture Notes in Math. 1448 (Springer-Verlag, 1990), 125–138.Google Scholar
4.Birkenmeier, G. F., Heatherly, H. E. and Lee, E. K., Completely prime ideal and associated radicals, Proceedings of Biennial Ohio State-Denison Conference 1992, (World Scientific, 1993), 102–129.Google Scholar
5.Birkenmeier, G. F., Kim, J. Y. and Park, J. K., A connection between weak regularity and the simplicity of prime factor rings, Proc. Amer. Math. Soc. 122 (1994), 53–58.CrossRefGoogle Scholar
6.Birkenmeier, G. F., Kim, J. Y. and Park, J. K., Regularity conditions and the simplicity of prime factor rings, J. Pure Appl. Algebra 115 (1997), 213–230.CrossRefGoogle Scholar
7.Fisher, J. W. and Snider, R. L., On the von Neumann regularity of rings with prime factor rings, Pacific J. Math. 54 (1974), 135–144.CrossRefGoogle Scholar
8.Hirano, Y., Some studies on strongly π-regular rings, Math. J. Okayama Univ. 20 (1978), 141–149.Google Scholar
9.Hirano, Y., Huynh, D. V. and Park, J. K., On rings whose prime radical contains all nilpotent elements of index two, Arch. Math. 66 (1996), 360–365.CrossRefGoogle Scholar
10.Hofmann, K. H., Representation of algebras by continuous sections, Bull. Amer. Math. Soc. 78 1972), 291–373.CrossRefGoogle Scholar
11.Kist, J., Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. 13 (1963), 31–50.CrossRefGoogle Scholar
12.Koh, K., On functional representations of a ring without nilpotent elements, Canad. Math. Bull. 14 (1971), 349–352.CrossRefGoogle Scholar
13.Koh, K., On a representation of a strongly harmonic ring by sheaves, Pacific J. Math. 41 (1972), 459–468.CrossRefGoogle Scholar
14.Lambek, J., On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359–368.CrossRefGoogle Scholar
15.Putcha, M. and Yaqub, A., Semigroups satisfying permutation identities, Semigroup Forum 3 (1971), 68–73.CrossRefGoogle Scholar
16.Putcha, M. and Yaqub, A., Rings satisfying monomial identities, Proc. Amer. Math. Soc. 32 (1972), 52–56.Google Scholar
17.Shin, G., Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43–60.CrossRefGoogle Scholar
18.Stewart, P. N., Semi-simple radical classes, Pacific J. Math. 32 (1970), 249–255.CrossRefGoogle Scholar
19.Sun, S.-H., Noncommutative rings in which every prime ideal is contained in a unique maximal ideal, J. Pure Appl. Algebra 76 (1991), 179–192.CrossRefGoogle Scholar
20.Sun, S.-H., A unification of some sheaf representations of rings by quotient rings, Comm. Algebra 22 (2) (1994), 687–696.CrossRefGoogle Scholar