1. CMB 2000 (vol 43 pp. 312)
 Dobbs, David E.

On the Prime Ideals in a Commutative Ring
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.
Categories:13C15, 13B25, 04A10, 14A05, 13M05 
