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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Commutative ideal theory without finiteness conditions: Completely irreducible ideals


Authors: Laszlo Fuchs, William Heinzer and Bruce Olberding
Journal: Trans. Amer. Math. Soc. 358 (2006), 3113-3131
MSC (2000): Primary 13A15, 13F05
Published electronically: March 1, 2006
MathSciNet review: 2216261
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Abstract: An ideal of a ring is completely irreducible if it is not the intersection of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without finiteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that admit a representation as an irredundant intersection of completely irreducible ideals, and we study the question of uniqueness of such representations. We characterize those commutative rings in which every ideal is an irredundant intersection of completely irreducible ideals.


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Additional Information

Laszlo Fuchs
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
Email: fuchs@tulane.edu

William Heinzer
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: heinzer@math.purdue.edu

Bruce Olberding
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-8001
Email: olberdin@emmy.nmsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03815-3
PII: S 0002-9947(06)03815-3
Keywords: Irreducible ideal, completely irreducible ideal, irredundant intersection, arithmetical ring
Received by editor(s): December 23, 2003
Received by editor(s) in revised form: June 4, 2004, and July 26, 2004
Published electronically: March 1, 2006
Article copyright: © Copyright 2006 American Mathematical Society