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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: [email protected]
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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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References

1.Bastida, E. and Gilmer, R., Overlings and divisorial ideals of rings of the form D + M, Michigan Math. J., 20 (1973), 79–95.CrossRefGoogle Scholar
2.Bedoya, H. and Lewin, J.. Ranks of matrices over Ore domains, Proc. Amer. Math. Soc., 62 (1977). 233–236.CrossRefGoogle Scholar
3.Bergman, G. M.. Commuting Elements in Free Algebras and Related Topics in Ring Theory, Ph.D. Thesis. Harvard University (Boston MA, 1967).Google Scholar
4.Brown, W. C., Null ideals and spanning ranks of matrices. Communications in Alg., 26 (1998), 2401–2417.CrossRefGoogle Scholar
5.Corm, P. M., Bezout rings and their subrings. Proc. Camb. Philos. Soc., 64 (1968). 251–264.Google Scholar
6.Colin, P. M., Unique factorization domains. Amer. Math. Monthly, 80 (1973), 1–18.Google Scholar
7.Cohn, P. M., Free Rings and Their Relations (2nd Ed.) Academic Press Inc., (London, 1985).Google Scholar
8.Dicks, W. and Sontag, E., Sylvester domains, J. Pure Appl. Alg., 13 (1978), 243–275.CrossRefGoogle Scholar
9.Fontana, M. and Gabelli, S., On the class group and local class group of a pullback. J. Alg., 181 (1996), 803–835.CrossRefGoogle Scholar
10.Fuchs, L., Riesz groups, Ann. Scoula Norm. Sup. Pisa, 19 (1965), 1–34.Google Scholar
11.Gabelli, S. and Houston, E., Coherentlike conditions in pullbacks, Michigan Math. J., 44 (1996). 99–123.Google Scholar
12.Gilmer, R., Multiplicative Ideal Theory, Queen's Papers in Pure and Applied Mathematics. Queen's University (Kingston, Ontario, 1992).Google Scholar
13.Heinzer, W. and Huneke, C., The Dedekind-Mertens lemma and the contents of polynomials. Proc. Amer. Math. Soc., 126 (1998), 1305–1309.CrossRefGoogle Scholar
14.Matlis, E., Torsion-free Modules, The University of Chicago Press (Chicago, 1972).Google Scholar
15.McAdam, S. and Rush, D. E., Schreier rings, Bull. London Math. Soc., 10 (1978), 77–80.CrossRefGoogle Scholar
16.Olberding, B., Stability of ideals and applications, Ideal Theoretic Methods in Commutative Algebra (Columbia, MO, 1999), 319–341, Lecture Notes in Pure and Appl. Math.. 220. Dekker (New York, 2001).Google Scholar
17.Olberding, B., Stability, duality, 2-generated ideals and canonical decomposition of modules. Rend. Sem. Mat. Univ. Padova, 106 (2001), 261–290.Google Scholar
18.Rush, D. E., Rings with two-generated ideals, J. Pure Appl. Alg., 73 (1991). 257–275.CrossRefGoogle Scholar
19.Rush, D. E., Two-generated ideals and representations of abelian groups over valuation rings. J. Alg., 177 (1995), 77–101.CrossRefGoogle Scholar
20.Rush, D. E., The Dedekind-Mertens lemma and the contents of polynomials, Proc. Amer. Math. Soc., 126 (1998), 2879–2884.Google Scholar
21.Teller, J. R., On partially ordered groups satisfying the Riesz interpolation property. Proc Amer. Math. Soc., 16 (1965), 1392–1400.CrossRefGoogle Scholar
22.Waterhouse, W. C., Quadratic polynomials and unique factorization, Amer. Math. Monthly, 109 (2002), 70–72.CrossRefGoogle Scholar
23.Zafrullah, M., On a property of pre-Schreier rings, Communications in Alg., 15 (1987). 1895–1920.CrossRefGoogle Scholar
24.Zafrullah, M., On Riesz groups, Manuscripta Math., 80 (1993), 225–238.CrossRefGoogle Scholar