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On a Generalization of the Euclidean Algorithm

Published online by Cambridge University Press:  24 October 2008

P. M. Cohn
P. M. Cohn
Affiliation:
The UniversityManchester 13
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Extract

In any valuated ring R, more specifically, a ring with a degree-function d(a) as defined in §2, the notion of right R-dependence may be denned as follows: (i) The elements a1, …, ar of R are right R-dependent if there exist b1, …, br ∈ R such that . (ii) An element a of R is said to be right R-dependent on the elements a1, …, ar of R if a = 0 or if there exist c1, …, cr ∈ R such that .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 1 , January 1961 , pp. 18 - 30
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

REFERENCES

(1)Birkhoff, G., Representability of Lie algebras and Lie groups by matrices. Ann. Math., Princeton, 38 (1937), 526–32.CrossRefGoogle Scholar
(2)Cohn, P. M., On the free product of associative rings. II. Math. Z. 73 (1960), 433–56.Google Scholar
(3)Levi, Howard,. A characterization of polynomial rings by means of order relations. Amer. J. Math. 65 (1943), 221–34.CrossRefGoogle Scholar
(4)McCoy, N. H., Rings and ideals (Baltimore, 1948).Google Scholar
(5)Ore, O., Linear equations in non-commutative fields. Ann. Math., Princeton, 32 (1931), 463–77.CrossRefGoogle Scholar
(6)Ore, O., Theory of non-commutative polynomials. Ann. Math., Princeton, 34 (1933), 480–508.CrossRefGoogle Scholar
(7)Schilling, O. F. G., The theory of valuations (New York, 1950).CrossRefGoogle Scholar
(8)Witt, E., Treue Darstellung Liescher Ringe. J. reine angew. Math. 177 (1937), 152–60.CrossRefGoogle Scholar