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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
MathSciNet review: 1291767
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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 $R$ of Goldie dimension 1 has Artinian classical quotient ring $Q$, hence is a Kerr ring in the sense that the polynomial ring $R[X]$ satisfies the $acc$ on annihilators $(=acc \bot )$. More generally, we show that a Goldie ring $R$ has Artinian $Q$ when every zero divisor of $R$ has essential annihilator (in this case $Q$ is a local ring; see Theorem $1^\prime $). A corollary to the proof is Theorem 2: A commutative ring $R$ has Artinian $Q$ iff $R$ is a Goldie ring in which each element of the Jacobson radical of $Q$ has essential annihilator. Applying a theorem of Beck we show that any $acc \bot $ ring $R$ that has Noetherian local ring $R_p$ for each associated prime $P$ is a Kerr ring and has Kerr polynomial ring $R[X]$ (Theorem 5).


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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: http://dx.doi.org/10.1090/S0002-9939-96-03028-6
PII: S 0002-9939(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