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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Polynomial rings over Goldie-Kerr commutative rings


Author: Carl Faith
Journal: Proc. Amer. Math. Soc. 120 (1994), 989-993
MSC: Primary 13E10; Secondary 13B25, 16P60
MathSciNet review: 1221723
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Abstract: All rings in this paper are commutative, and $ \operatorname{acc} \bot $ (resp., $ \operatorname{acc} \, \oplus $) denotes the acc on annihilators (resp., on direct sums of ideals). Any subring of an $ \operatorname{acc} \bot $ ring, e.g., of a Noetherian ring, is an $ \operatorname{acc} \bot $ ring. Together, $ \operatorname{acc} \bot $ and $ \operatorname{acc} \, \oplus $ constitute the requirement for a ring to be a Goldie ring. Moreover, a ring $ R$ is Goldie iff its classical quotient ring $ Q$ is Goldie.

A ring $ R$ is a Kerr ring (the appellation is for J. Kerr, who in 1990 constructed the first Goldie rings not Kerr) iff the polynomial ring $ R[x]$ has $ \operatorname{acc} \bot $ (in which case $ R$ must have $ \operatorname{acc} \bot $). By the Hilbert Basis theorem, if $ S$ is a Noetherian ring, then so is $ S[x]$; hence, any subring $ R$ of a Noetherian ring is Kerr.

In this note, using results of Levitzki, Herstein, Small, and the author, we show that any Goldie ring $ R$ such that $ Q = {Q_c}(R)$ has nil Jacobson radical (equivalently, the nil radical of $ R$ is an intersection of associated prime ideals) is Kerr in a very strong sense: $ Q$ is Artinian and, hence, Noetherian (Theorems 1.1 and 2.2). As a corollary we prove that any Goldie ring $ A$ that is algebraic over a field $ k$ is Artinian, and, hence, any order $ R$ in $ A$ is a Kerr ring (Theorem 2.5 and Corollary 2.6). The same is true of any algebra $ A$ over a field $ k$ of cardinality exceeding the dimension of $ A$ (Corollary 2.7).

Other Kerr rings are: reduced $ \operatorname{acc} \bot $ rings and valuation rings with $ \operatorname{acc} \bot $ (see 3.3 and 3.4).


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DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1221723-8
PII: S 0002-9939(1994)1221723-8
Article copyright: © Copyright 1994 American Mathematical Society