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The ideals of the hurwitzean polynomial ring

Published online by Cambridge University Press:  17 April 2009

Margaret J. Morton
Margaret J. Morton
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania, USA.
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In 1919, Adolf Hurwitz formed the quaternion ring R composed of elements whose coordinates were either all integers or halves of odd integers. The objective of this paper is to examine the (two-sided) ideal structure in the hurwitzean polynomial ring R[x], formed by taking all polynomials with coefficients in R. The maximal and prime ideals of R[x] will be characterized with results surprisingly analogous to those in Z[x]. In addition, a canonical basis, of the type developed by G. Szekeres, 1952, for polynomial domains, will be developed for the ideals of R[x].

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 1 , February 1976 , pp. 37 - 61
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Hurwitz, Adolf, Vorlesungen über die Zahlentheone der Quaternionen (Julius Springer, Berlin, 1919).CrossRefGoogle Scholar
[2]Redéi, L., Algebra, Volume 1 (International Series of Monographs in Pure and Applied Mathematics, 91. Pergamon, Oxford, London, Edinburgh, New York, Toronto, Sydney, Paris, Braunschweig; Akadémiai Kiadó, Hungary; 1967).Google Scholar
[3]Szekeres, G., “A canonical basis for the ideals of a polynomial domain”, Amer. Math. Monthly 59 (1952), 379–386.CrossRefGoogle Scholar