Contracted ideals and purity for ring extensions

Abstract:

In this paper an example is given of a pair of commutative noetherian rings $R \subseteq S$ with $S$ a finite $R$-module and $IS \cap R = I$ for each ideal $I$ of $R$, but having the property that $0 \to R \to S$ is not a pure sequence of $R$-modules. Purity of the sequence $0 \to R \to S$ is equivalent to $R[X]$ being “ideally closed” in $S[X],\;X$ an indeterminate. Therefore, the example renders appealing the proposition that for $R$ noetherian and $S$ a noetherian torsion-free $R$-algebra containing $R$, if $\alpha S \cap R = \alpha R$ for each non-zero-divisor $\alpha \epsilon R$, then the extension $R[X] \subseteq S[X]$ has the same properties. Finally, it is also shown that for $R$ noetherian and $0 \to R \to S$ pure, with $S$ an $R$-algebra, then $R[[{X_1}, \ldots ,{X_n}]]$ is pure in $S[[{X_1}, \ldots ,{X_n}]]$ for each positive integer $n$.