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 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 for each . Theorem 2 gives the analogous result for going down in 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.

**[1]**Stephen McAdam, Private communication.**[2]**Claudio Procesi,*Rings with polynomial identities*, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, 17. MR**0366968****[3]**William Schelter,*Integral extensions of rings satisfying a polynomial identity*, J. Algebra**40**(1976), no. 1, 245–257. MR**0417238****[4]**-,*Non-commutative affine*P.I.*rings are Catenary*(to appear).**[5]**Louis Rowen,*Some results on the center of a ring with polynomial identity*, Bull. Amer. Math. Soc.**79**(1973), 219–223. MR**0309996**, 10.1090/S0002-9904-1973-13162-3**[6]**I. S. Cohen and A. Seidenberg,*Prime ideals and integral dependence*, Bull. Amer. Math. Soc.**52**(1946), 252–261. MR**0015379**, 10.1090/S0002-9904-1946-08552-3

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DOI:
https://doi.org/10.1090/S0002-9939-1977-0432619-3

Keywords:
PI ring,
going down,
simple going down,
integral,
extension,
going up,
lying over,
incomparability

Article copyright:
© Copyright 1977
American Mathematical Society