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A Remark on Coherent Overrings

Published online by Cambridge University Press:  20 November 2018

Ira J. Papick
Ira J. Papick*
Affiliation:
Adelphi University, Garden City,New York 11530
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Throughout this note, let R be a (commutative integral) domain with quotient field K. A domain S satisfying R ⊆ S ⊆ K is called an overring of R, and by dimension of a ring we mean Krull dimension. Recall [1] that a commutative ring is said to be coherent if each finitely generated ideal is finitely presented.

In [2], as a corollary of a more general theorem, Davis showed that if each overring of a domain R is Noetherian, then the dimension of R is at most 1. (This corollary is the converse of a version of the Krull-Akizuki Theorem [5, Theorem 93], and can also be proved directly by using the existence of valuation rings dominating finite chains of prime ideals [4, Corollary 16.6].) It is our purpose to prove that if R is Noetherian and each overring of R is coherent, then the dimension of £ is at most 1. We shall also indicate some related questions and examples.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 3 , 01 September 1978 , pp. 373 - 375
Copyright
Copyright © Canadian Mathematical Society 1978

References

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