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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, 13B25, 04A10, 14A05, 13M05 show english descriptions Dimension theory, depth, related rings (catenary, etc.)
Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]
unknown classification 04A10
Relevant commutative algebra [See also 13-XX]
Structure
13C15 - Dimension theory, depth, related rings (catenary, etc.)
13B25 - Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]
04A10 - unknown classification 04A10
14A05 - Relevant commutative algebra [See also 13-XX]
13M05 - Structure
 

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