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

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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 $R$ 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

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