Bond invariance of 𝐺-rings and localization

Warfield, Robert B.

doi:10.1090/S0002-9939-1991-1027102-8

Bond invariance of $G$-rings and localization
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Proc. Amer. Math. Soc. 111 (1991), 13-18 Request permission

Abstract:

It is proved that if $R$ and $S$ are prime Noetherian rings and there exists an $(R,S)$-bimodule that is finitely generated and torsionfree on each side, then the intersection of the nonzero prime ideals of $R$ is nonzero if and only if the same holds for the corresponding intersection in $S$. Consequently, if the right primitive ideals in a given Noetherian ring are precisely the locally closed prime ideals, then the same equivalence holds true for any finite extension ring. Another consequence of the methods used here is the following answer to a question of Braun: If the intersection of the prime ideals in a clique in a Noetherian PI ring is a prime ideal $Q$, then $Q$ is localizable.

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