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1. CMB 2000 (vol 43 pp. 312)
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
best-possible.
Categories:13C15, 13B25, 04A10, 14A05, 13M05 |
2. CMB 1999 (vol 42 pp. 231)
Generating Ideals in Rings of Integer-Valued Polynomials Let $R$ be a one-dimensional locally analytically irreducible
Noetherian domain with finite residue fields. In this note it is
shown that if $I$ is a finitely generated ideal of the ring
$\Int(R)$ of integer-valued polynomials such that for each $x \in
R$ the ideal $I(x) =\{f(x) \mid f \in I\}$ is strongly
$n$-generated, $n \geq 2$, then $I$ is $n$-generated, and some
variations of this result.
Categories:13B25, 13F20, 13F05 |