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Quadratic polynomials, factorization in integral domains and Schreier domains from pullbacks

Part of: General commutative ring theory

Published online by Cambridge University Press:  26 February 2010

David E. Rush
David E. Rush
Affiliation:
Department of Mathematics, University of California, Riverside 92521, U.S.A., E-mail: rush@math.ucr.edu
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Abstract

Let R be an integral domain with quotient field K and let X be an indeterminate. A result of W. C. Waterhouse states that, if each quadratic polynomial f ∈ R[X] which factors into linear polynomials in K[X] also factors into linear polynomials in R[X], then every irreducible element in R is prime. In this note the rings which satisfy the hypothesis of this theorem are characterized, and compared to the rings for which each polynomial f ∈ R[X] which factors into two polynomials of positive degree in K[X] also factors into two polynomials of positive degree in R[X]. Relevant examples are furnished via the pullback construction.

MSC classification

Secondary: 13A05: Divisibility; factorizations
Type
Research Article
Information
Mathematika , Volume 50 , Issue 1-2 , December 2003 , pp. 103 - 112
Copyright
Copyright © University College London 2003

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