JP Journal of Algebra, Number Theory and Applications
Volume 16, Issue 1, Pages 41 - 61
(February 2010)
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RINGS WHOSE CENTERS HAVE A COMMON IDEAL
Deborah Kaplan (Polakiewicz)
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Abstract: Much of algebraic geometry is based on chains of prime ideals in commutative noetherian rings, in particular affine rings.
Now if a ring Cis not noetherian (or affine) but it has a common ideal with an overring Rwhich is noetherian (or affine), then chains of prime ideals in Ccan be lifted to chains of prime ideals in R, and under some additional conditions, some properties of Rlike cattenarity or finite Krull dimension or classical Krull dimension are transmitted to C.
The same thing can occur when the ring Cstarts a chain of rings � �where every pair of consecutive rings has a common ideal.
The study of such a case for commutative rings is developed in the paper: �ring-theoretic properties of commutative algebras of invariants� of Kantor and Rowen [12]. They define such a ring as a �nearly noetherian� or �nearly affine ring�, or in general they give the definition of �nearly C� rings where C is a specific class of rings and the last component of the chain is in C.
For the noncommutative rings I have developed some results in the paper �very nearly C� rings [13].
The purpose of this paper is to study those properties for the case of chains of rings where the centers of two consecutive components of the chain have a common ideal. |
Keywords and phrases: common ideal, PI rings, nearly noetherian rings, nearly affine rings. |
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