Abstract view
On the Prime Ideals in a Commutative Ring


Published:20000901
Printed: Sep 2000
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
bestpossible.
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 13XX] 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 13XX] 13M05  Structure
