## Polynomial rings over Goldie-Kerr commutative rings II

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- by Carl Faith PDF
- Proc. Amer. Math. Soc.
**124**(1996), 341-344 Request permission

## 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
- Communicated by: Wolmer V. Vasconcelos
- © Copyright 1996 American Mathematical Society
- 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

Dedicated: In memory of Pere Menal