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Free and Injective Lie Modules*

Published online by Cambridge University Press:  20 November 2018

Israel Kleiner
Israel Kleiner*
Affiliation:
McGill University
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We study free and injective Lie modules by investigating the relationship between Lie modules and (associative) modules. An important role is played by the universal enveloping ring of a Lie ring [4]. If L is an arbitrary Lie ring and W(L) its universal enveloping ring, we show that the category of Lie L-modules and the category of associative W(L)-module s are isomorphic (section 2). In section 3 we study free Lie modules and show how they may be obtained from free associative modules. A Lie module is free if and only if it is a direct sum of copies of the free Lie module on one generator.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 9 , Issue 1 , April 1966 , pp. 29 - 42
Copyright
Copyright © Canadian Mathematical Society 1966

Footnotes

*

I wish to thank Professor J, Lambek for many helpful suggestions.

References

1. Jacobson, N., Lie Algebras, Interscience, New York, 1962.Google Scholar
2. Lambek, J., Lectures on Rings and Modules, Blaisdell, New York, 1966.Google Scholar
3. MacLane, S., Categorical Algebra, Bull. Amer. Math. Soc. 71 (1965), pp. 40-106.Google Scholar
4. Witt, E., Treue Darstellungen Beliebiger Liescher Ringe, Collect. Math. 6 (1953), pp. 107-114.Google Scholar