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On the Prime Ideals in a Commutative Ring

Published online by Cambridge University Press:  20 November 2018

David E. Dobbs
David E. Dobbs*
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA
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Abstract

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If $n$ and $m$ are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring $R$ with exactly $n$ elements and exactly $m$ prime ideals. Next, assuming the Axiom of Choice, it is proved that if $R$ is a commutative ring and $T$ is a commutative $R$ -algebra which is generated by a set $I$ , then each chain of prime ideals of $T$ lying over the same prime ideal of $R$ has at most ${{2}^{\left| I \right|}}$ elements. A polynomial ring example shows that the preceding result is best-possible.

Keywords

13C1513B2504A1014A0513M05

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 43 , Issue 3 , 01 September 2000 , pp. 312 - 319
Copyright
Copyright © Canadian Mathematical Society 2000

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