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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Simple going down in PI rings

Author: Phillip Lestmann
Journal: Proc. Amer. Math. Soc. 63 (1977), 41-45
MSC: Primary 13A15; Secondary 13B99, 13F10
MathSciNet review: 0432619
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Abstract: In this paper we prove two generalizations of a theorem which McAdam proved for commutative rings. Theorem 1 states that if $ R \subset S$ is a central integral extension of PI rings, then going down for prime ideals holds between R and S if and only if going down holds in $ R \subset R[s]$ for each $ s \in S$. Theorem 2 gives the analogous result for going down in $ C \subset R$ where C is a central subring of the PI ring R. As a corollary we obtain a result of Schelter generalizing Krull's theorem on going down for integral extensions of integrally-closed subrings.

References [Enhancements On Off] (What's this?)

  • [1] Stephen McAdam, Private communication.
  • [2] Claudio Procesi, Rings with polynomial identities, Dekker, New York, 1973. MR 51 #3214. MR 0366968 (51:3214)
  • [3] William Schelter, Integral extensions of rings satisfying a polynomial identity, J. Algebra 40 (1976), 245-257. MR 0417238 (54:5295)
  • [4] -, Non-commutative affine P.I. rings are Catenary (to appear).
  • [5] L. Rowen, Some results on the center of a ring with polynomial identity, Bull. Amer. Math. Soc. 79 (1973), 219-223. MR 46 #9099. MR 0309996 (46:9099)
  • [6] I. Cohen and A. Seidenberg, Prime ideals and integral dependence, Bull. Amer. Math. Soc. 52 (1946), 252-261. MR 7, 410. MR 0015379 (7:410a)

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Keywords: PI ring, going down, simple going down, integral, extension, going up, lying over, incomparability
Article copyright: © Copyright 1977 American Mathematical Society

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