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Rings of Quotients of Rings of Derivations

Published online by Cambridge University Press:  20 November 2018

Israel Kleiner
Israel Kleiner*
Affiliation:
York University Toronto
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The concept of a rational extension of a Lie module is defined as in the associative case [1, pp. 81 and 79]. It then follows from [3, Theorem 2.3] that any Lie module possesses a maximal rational extension (a rational completion), unique up to isomorphism. If now L and K are Lie rings with L⊆ K, we call K a (Lie) ring of quotients of L if K, considered as a Lie module over L, is a rational extension of the Lie module LL. Although we do not know if for every Lie ring L its rational completion can be given a Lie ring structure extending that of L (as is the case for associative rings), this is so, in any case, for abelian Lie rings (Propositions 2 and 4).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 3 , August 1968 , pp. 383 - 398
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Findlay, G.D. and Lambek, J., A generalized ring of quotients I, II. Can. Math. Bull. 1 (1958)77-85, 155–167.Google Scholar
2. Jacobson, N., Lie algebras. Inter science, New York, 1962.Google Scholar
3. Kleiner, I., Free and injective Lie modules. Can. Math. Bull. 9 (1966) 29-42.Google Scholar
4. Lambek, J., Lectures on rings and modules. Blaisdell, New York, 1966.Google Scholar
5. Tewari, K., Complexes over a complete algebra of quotients. Can, J. Math. 19 (1967) 40-57.Google Scholar
6. Zariski, O. and Samuel, P., Commutative algebra, Vol. I. Van Nostrand, Princeton, 1958.Google Scholar