Simple going down in PI rings
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- by Phillip Lestmann PDF
- Proc. Amer. Math. Soc. 63 (1977), 41-45 Request permission
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
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Stephen McAdam, Private communication.
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- William Schelter, Integral extensions of rings satisfying a polynomial identity, J. Algebra 40 (1976), no. 1, 245–257. MR 417238, DOI 10.1016/0021-8693(76)90095-8 —, Non-commutative affine P.I. rings are Catenary (to appear).
- Louis Rowen, Some results on the center of a ring with polynomial identity, Bull. Amer. Math. Soc. 79 (1973), 219–223. MR 309996, DOI 10.1090/S0002-9904-1973-13162-3
- I. S. Cohen and A. Seidenberg, Prime ideals and integral dependence, Bull. Amer. Math. Soc. 52 (1946), 252–261. MR 15379, DOI 10.1090/S0002-9904-1946-08552-3
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 63 (1977), 41-45
- MSC: Primary 13A15; Secondary 13B99, 13F10
- DOI: https://doi.org/10.1090/S0002-9939-1977-0432619-3
- MathSciNet review: 0432619