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Simple rings without zero-divisors, and Lie division rings

Published online by Cambridge University Press:  26 February 2010

P. M. Cohn
P. M. Cohn
Affiliation:
The University, Manchester 13.
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Extract

Let R be a simple ring. If R contains at least one minimal nonzero one-sided ideal, then R has zero-divisors, unless R is a division ring. However, simple rings exist which are not division rings and have no zero-divisors. Our present object is to prove the following embedding theorem:

Theorem 1. Every ring R without zero-divisors may be embedded in a simple ring R* without zero-divisors. If there is a non-zero element ƒ in R satisfying ƒ2 = nƒ, where n is an integer, then R* necessarily has a unit-element; otherwise R* may be chosen to have no unit-element.

Type
Research Article
Information
Mathematika , Volume 6 , Issue 1 , June 1959 , pp. 14 - 18
Copyright
Copyright © University College London 1959

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References

1.Birkhoff, G., “Representability of Lie algebras and Lie groups by matrices”, Annals of Math., 38 (1937), 526–532.CrossRefGoogle Scholar
2.Cohn, P. M., “On a class of simple rings”, Mathematika, 5 (1958), 103–117.CrossRefGoogle Scholar
3.Jacobson, N., “Structure theory of simple rings without finiteness assumptions”, Trans. American Math. Soc., 57 (1945), 228–245.CrossRefGoogle Scholar
4.Neumann, B. H., “Embedding non-associative rings in division rings,”, Proc. London Math. Soc. (3), 1 (1951), 241–256.CrossRefGoogle Scholar
5.Tamari, D., “On the embedding of Birkhoff-Witt rings in quotient fields”, Proc. American Math. Soc., 4 (1953), 197–202.CrossRefGoogle Scholar
6.Witt, E., “Treue Darstellung Liescher Binge”, Journal für die reine u. angew. Math., 177 (1937), 152–160.CrossRefGoogle Scholar