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Proceedings of the American Mathematical Society

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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

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
  • © 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