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Polynomial rings over Goldie-Kerr commutative rings II |
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| Authors: | Carl Faith |
| Affiliation: | Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 - Permanent address: 199 Longview Drive, Princeton, New Jersey 08540 |
| 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  of Goldie dimension 1 has Artinian classical quotient ring  , hence is a Kerr ring in the sense that the polynomial ring ![$R[X]$](http://www.ams.org/proc/1996-124-02/S0002-9939-96-03028-6/gif-abstract/img4.gif) satisfies the  on annihilators  . More generally, we show that a Goldie ring  has Artinian  when every zero divisor of  has essential annihilator (in this case  is a local ring; see Theorem  ). A corollary to the proof is Theorem 2: A commutative ring  has Artinian  iff  is a Goldie ring in which each element of the Jacobson radical of  has essential annihilator. Applying a theorem of Beck we show that any  ring  that has Noetherian local ring  for each associated prime  is a Kerr ring and has Kerr polynomial ring ![$R[X]$](http://www.ams.org/proc/1996-124-02/S0002-9939-96-03028-6/gif-abstract/img20.gif) (Theorem 5). |
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