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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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


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
  • © 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:
  • MathSciNet review: 1291767