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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 | 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).

References [Enhancements On Off] (What's this?)

  • B I. Beck, $\Sigma $-injective modules, J. Algebra 21 (1972), 232--249, MR 50:9967.
  • C V. Camillo, Coherence for polynomial rings, J. Algebra 132 (1990), 72--76, MR 91c:16018.
  • C-H F. Cedó and D. Herbera, On polynomial rings over Kerr commutative rings, preprint, U. Autónoma de Barcelona, 1995.
  • F-F A. Facchini and C. Faith, FP-injective quotient rings and elementary divisor rings, Proceedings of the Fez Conference on Commutative Algebra (1995), Lecture Notes in Pure and Appl. Math., Marcel Dekker, New York and Basel, 1996.
  • F1 C. Faith, Finitely embedded commutative rings, Proc. Amer. Math. Soc. 112 (1991), 657--659, MR 93f:13012.
  • F2 ------, Polynomial rings over Goldie-Kerr commutative rings, Proc. Amer. Math. Soc. 120 (1994), 989--993, MR 94k:13024.
  • F3 ------, Algebra II: Ring theory, Springer-Verlag, Berlin, Heidelberg, and New York, 1976, MR 55:383.
  • F4 ------, Annihilators, associated primes and Kasch-McCoy quotient rings of commutative rings, Comm. Algebra 119 (1991), 1867--1892, MR 92g:16008.
  • F-P C. Faith and P. Pillay, Classification of commutative FPF rings, Notas Mat., vol. 4, Univ. Murcia, Murcia.
  • H J. Huckaba, Commutative rings with zero divisors, Monographs Pure Appl. Math., Marcel Dekker, Basel and New York, 1988, MR 89e:13001.
  • K1 J. W. Kerr, The polynomial ring over a Goldie ring need not be a Goldie ring, J. Algebra 134 (1990), 344--352, MR 91h:16042.
  • K2 ------, An example of a Goldie ring whose matrix ring is not Goldie, J. Algebra 61 (1979), 590--592, MR 81b:16016.
  • S L. Small, Orders in Artinian rings, J. Algebra 4 (1966), 13--41, MR 34:199.

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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

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