Contracted ideals and purity for ring extensions
J. W. Brewer and D. L. Costa
Proc. Amer. Math. Soc. 53 (1975), 271-276
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Abstract: In this paper an example is given of a pair of commutative noetherian rings with a finite -module and for each ideal of , but having the property that is not a pure sequence of -modules. Purity of the sequence is equivalent to being ``ideally closed'' in an indeterminate. Therefore, the example renders appealing the proposition that for noetherian and a noetherian torsion-free -algebra containing , if for each non-zero-divisor , then the extension has the same properties. Finally, it is also shown that for noetherian and pure, with an -algebra, then is pure in for each positive integer .
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