Bond invariance of -rings and localization

Author:
Robert B. Warfield

Journal:
Proc. Amer. Math. Soc. **111** (1991), 13-18

MSC:
Primary 16N60; Secondary 16D20, 16D30, 16D60, 16P50

DOI:
https://doi.org/10.1090/S0002-9939-1991-1027102-8

MathSciNet review:
1027102

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Abstract: It is proved that if and are prime Noetherian rings and there exists an -bimodule that is finitely generated and torsionfree on each side, then the intersection of the nonzero prime ideals of is nonzero if and only if the same holds for the corresponding intersection in . 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 , then is localizable.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1027102-8

Keywords:
Noetherian ring,
Noetherian bimodule,
bond,
prime ideal,
primitive ideal,
-ring,
locally closed,
ring extension,
clique,
localization,
PI ring

Article copyright:
© Copyright 1991
American Mathematical Society