Journal article
The u,u−1 lemma revisited
Communications in Algebra, Vol.24(7), pp.2447-2454
01/01/1996
DOI: 10.1080/00927879608825708
Abstract
Let it be an integral domain with quotient field K. The u,u −1 Lemma states that if R is integrally closed and quasilocal and if u ∈ K is the root of a polynomial f ∑ R [X] with some coefficient a unit, then u or u −1 ∈ R. A globalization states that for R integrally closed, if is the root of f ∈ R [X] with A f invertible, then (a, 6) is invertible. We prove the converse of both results and show that for R integrally closed, the following are equivalent: (1) R is Prüfer, (2) every u ∑ K is the root of a quadratic polynomial f ∑ R [X] with some coefficient a unit, and (3) every u ∈ K is the root of a polynomial f ∈ R [X] with A f invertible. Moreover, for any integral domain R, the integral closure is Prüfer if and only if (3) holds.
Details
- Title: Subtitle
- The u,u−1 lemma revisited
- Creators
- D. D Anderson - Department of Mathematics , The University of IowaDong Je Kwak - Department of Mathematics , Kyungpook National University
- Resource Type
- Journal article
- Publication Details
- Communications in Algebra, Vol.24(7), pp.2447-2454
- Publisher
- Marcel Dekker, Inc
- DOI
- 10.1080/00927879608825708
- ISSN
- 0092-7872
- eISSN
- 1532-4125
- Language
- English
- Date published
- 01/01/1996
- Academic Unit
- Mathematics
- Record Identifier
- 9983985873102771
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