Unibranched prime ideals and going down in PI rings

Author:
Phillip Lestmann

Journal:
Proc. Amer. Math. Soc. **82** (1981), 191-195

MSC:
Primary 16A33; Secondary 16A38

DOI:
https://doi.org/10.1090/S0002-9939-1981-0609649-2

MathSciNet review:
609649

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Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to answer the question of whether going down is equivalent to unibranchedness of prime ideals in integral extensions of prime rings. We show by example that, in general, the answer is no; and we find an additional condition which, together with going down, implies prime ideals of are unibranched.

**[1]**S. A. Amitsur and L. Small,*Prime**-rings*, Bull. Amer. Math. Soc.**83**(1977), 249-251. MR**0435134 (55:8095)****[2]**V. A. Jategaonkar,*Principal ideal theorem for Noetherian**rings*, J. Algebra**35**(1975), 17-22. MR**0371944 (51:8161)****[3]**Phillip Lestmann,*Going down and the spec map in**rings*, Comm. Algebra**6(16)**(1978), 1667-1691. MR**508243 (80a:16032)****[4]**Stephen McAdam,*Going down*, Duke Math. J.**39**(1972), 633-636. MR**0311658 (47:220)****[5]**Neal H. McCoy,*A note on finite unions of ideals and subgroups*, Proc. Amer. Math. Soc.**8**(1957), 633-637. MR**0086803 (19:246b)****[6]**Claudio Procesi,*Rings with polynomial identities*, Dekker, New York, 1973. MR**0366968 (51:3214)****[7]**William Schelter,*Integral extensions of rings satisfying a polynomial identity*, J. Algebra**40**(1976), 245-257. MR**0417238 (54:5295)**

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DOI:
https://doi.org/10.1090/S0002-9939-1981-0609649-2

Keywords:
Integral extension,
integral closure,
going down,
prime ideal,
ring,
Noetherian,
unibranched

Article copyright:
© Copyright 1981
American Mathematical Society