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 (resp., ) denotes the acc on annihilators (resp., on direct sums of ideals). Any subring of an ring, e.g., of a Noetherian ring, is an ring. Together, and constitute the requirement for a ring to be a Goldie ring. Moreover, a ring is Goldie iff its classical quotient ring is Goldie.

A ring is a Kerr ring (the appellation is for J. Kerr, who in 1990 constructed the first Goldie rings not Kerr) iff the polynomial ring has (in which case must have ). By the Hilbert Basis theorem, if is a Noetherian ring, then so is ; hence, any subring of a Noetherian ring is Kerr.

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

Other Kerr rings are: reduced rings and valuation rings with (see 3.3 and 3.4).

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DOI:
https://doi.org/10.1090/S0002-9939-1994-1221723-8

Article copyright:
© Copyright 1994
American Mathematical Society