Intermediate normalizing extensions

Authors:
A. G. Heinicke and J. C. Robson

Journal:
Trans. Amer. Math. Soc. **282** (1984), 645-667

MSC:
Primary 16A56; Secondary 16A26, 16A34, 16A55, 16A66

DOI:
https://doi.org/10.1090/S0002-9947-1984-0732112-2

MathSciNet review:
732112

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Abstract: Relationships between the prime ideals of a ring and of a normalizing extension have been studied by several authors recently. In this work, most of these known results are extended to give relationships between the prime ideals of and of where is a ring with , and is a normalizing extension of : such rings are called *intermediate normalizing extensions* of .

One result ("Cutting Down") asserts that for any prime ideal of , is the intersection of a finite set of prime ideals of , uniquely defined by , whose corresponding factor rings are mutually isomorphic. The minimal members of the family of 's are the primes of minimal over , and an "incomparability" theorem is proved which shows that no two comparable primes of can have their intersections with share a common minimal prime. Other results include versions of the "lying over" and "going up" theorems, proofs that chain conditions such as right Goldie or right Noetherian pass between and each of the rings , and a demonstration that the "additivity principle" holds.

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DOI:
https://doi.org/10.1090/S0002-9947-1984-0732112-2

Article copyright:
© Copyright 1984
American Mathematical Society