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Some finitely based varieties of rings

Published online by Cambridge University Press:  09 April 2009

Trevor Evans
Trevor Evans
Affiliation:
Emory University, Atlanta, Georgia, 30322, U.S.A.
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The results in this paper are consequences of an attempt many years ago to extend to loops some form of the theorem of Lyndon [12] that any nilpotent group has finitely based identities. Having failed in this, we looked for other algebras for which a similar approach might work. The algebra has to belong to a variety in which finitely generated algebras are finitely related and we must be able to bound the number of variables needed in a basis. Commutative Moufang loops, because of the extensive commutator calculus available (Bruck, [4]), provide one example (Evans, [6]). Here we give two examples from rings, namely associative rings satisfying xn = x (more generally, satisfying an identity x2 · p(x) = x) and nilpotent (non-associative) rings. We are also able to extend some results of Higman [9] on product varieties and we show that for associative rings the product of a nilpotent variety and a finitely based bariety is finitely based.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 17 , Issue 2 , March 1974 , pp. 246 - 255
Copyright
Copyright © Australian Mathematical Society 1974

References

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