Simple going down in PI rings

Author:
Phillip Lestmann

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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*, 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)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
13A15,
13B99,
13F10

Retrieve articles in all journals with MSC: 13A15, 13B99, 13F10

Additional Information

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