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# On the Prime Ideals in a Commutative Ring

Published:2000-09-01
Printed: Sep 2000
• David E. Dobbs
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## Abstract

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 $2^{|I|}$ elements. A polynomial ring example shows that the preceding result is best-possible.
 MSC Classifications: 13C15 - Dimension theory, depth, related rings (catenary, etc.) 13B25 - Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] 04A10 - unknown classification 04A1014A05 - Relevant commutative algebra [See also 13-XX] 13M05 - Structure