Polynomial rings

over Goldie-Kerr commutative rings II

Author:
Carl Faith

Journal:
Proc. Amer. Math. Soc. **124** (1996), 341-344

MSC (1991):
Primary 13B25, 13CO5, 13EO5, 13H99, 13J10; Secondary 16D90, 16P60, 16S50

DOI:
https://doi.org/10.1090/S0002-9939-96-03028-6

MathSciNet review:
1291767

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An overlooked corollary to the main result of the stated paper (Proc. Amer. Math. Soc. **120** (1994), 989--993) is that any Goldie ring of Goldie dimension 1 has Artinian classical quotient ring , hence is a Kerr ring in the sense that the polynomial ring satisfies the on annihilators . More generally, we show that a Goldie ring has Artinian when every zero divisor of has essential annihilator (in this case is a local ring; see Theorem ). A corollary to the proof is Theorem 2: A commutative ring has Artinian iff is a Goldie ring in which each element of the Jacobson radical of has essential annihilator. Applying a theorem of Beck we show that any ring that has Noetherian local ring for each associated prime is a Kerr ring and has Kerr polynomial ring (Theorem 5).

**B**I. Beck,*-injective modules*, J. Algebra**21**(1972), 232--249, MR**50:9967**.**C**V. Camillo,*Coherence for polynomial rings*, J. Algebra**132**(1990), 72--76, MR**91c:16018**.**C-H**F. Cedó and D. Herbera,*On polynomial rings over Kerr commutative rings*, preprint, U. Autónoma de Barcelona, 1995.**F-F**A. Facchini and C. Faith,*FP-injective quotient rings and elementary divisor rings*, Proceedings of the Fez Conference on Commutative Algebra (1995), Lecture Notes in Pure and Appl. Math., Marcel Dekker, New York and Basel, 1996.**F1**C. Faith,*Finitely embedded commutative rings*, Proc. Amer. Math. Soc.**112**(1991), 657--659, MR**93f:13012**.**F2**------,*Polynomial rings over Goldie-Kerr commutative rings*, Proc. Amer. Math. Soc.**120**(1994), 989--993, MR**94k:13024**.**F3**------,*Algebra II: Ring theory*, Springer-Verlag, Berlin, Heidelberg, and New York, 1976, MR**55:383**.**F4**------,*Annihilators, associated primes and Kasch-McCoy quotient rings of commutative rings*, Comm. Algebra**119**(1991), 1867--1892, MR**92g:16008**.**F-P**C. Faith and P. Pillay,*Classification of commutative FPF rings*, Notas Mat., vol. 4, Univ. Murcia, Murcia.**H**J. Huckaba,*Commutative rings with zero divisors*, Monographs Pure Appl. Math., Marcel Dekker, Basel and New York, 1988, MR**89e:13001**.**K1**J. W. Kerr,*The polynomial ring over a Goldie ring need not be a Goldie ring*, J. Algebra**134**(1990), 344--352, MR**91h:16042**.**K2**------,*An example of a Goldie ring whose matrix ring is not Goldie*, J. Algebra**61**(1979), 590--592, MR**81b:16016**.**S**L. Small,*Orders in Artinian rings*, J. Algebra**4**(1966), 13--41, MR**34:199**.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
13B25,
13CO5,
13EO5,
13H99,
13J10,
16D90,
16P60,
16S50

Retrieve articles in all journals with MSC (1991): 13B25, 13CO5, 13EO5, 13H99, 13J10, 16D90, 16P60, 16S50

Additional Information

**Carl Faith**

Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903;
Permanent address: 199 Longview Drive, Princeton, New Jersey 08540

DOI:
https://doi.org/10.1090/S0002-9939-96-03028-6

Received by editor(s):
April 25, 1994

Received by editor(s) in revised form:
August 5, 1994

Dedicated:
In memory of Pere Menal

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 1996
American Mathematical Society