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Search: MSC category 13B25 ( Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] )

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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 best-possible.

Categories:13C15, 13B25, 04A10, 14A05, 13M05

2. CMB 1999 (vol 42 pp. 231)

Rush, David E.
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

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