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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

[1]Baker, K. A., ‘Equational bases for finite algebras’, (Preliminary report), Notices Amer. Math. Soc. 19 (1972), 691–08-2.Google Scholar
[2]Birkhoff, G., ‘On the structure of abstract algebras’, Proc. Cambr. Philos. Soc. 31 (1935), 433–454.CrossRefGoogle Scholar
[3]Bang, C. M. and Mandelberg, K., Finite basis theorems for rings satisfying a central condition (in preparation).Google Scholar
[4]Bruck, R. H., A Survey of Binary Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete vol. 20, (Springer Verlag, 1958).CrossRefGoogle Scholar
[5]Cohen, D. E., ‘On the laws of a metabelian variety’, J. of Alg. 5 (1967), 267–273.CrossRefGoogle Scholar
[6]Evans, T., ‘Identites and reaktions in commutative Moufang loops’, J. of Algebra (to appear).Google Scholar
[7]Evans, T., Identical relations in loops II (in preparation).Google Scholar
[8]Herstein, I. N., ‘The structure of a certain class of rings’, Amer. J. Math. 75 (1953), 864–871.CrossRefGoogle Scholar
[9]Higman, G., ‘Some remarks on varieties of groups’, Quart. J. Math. Oxford (2) 10 (1969), 165–178.Google Scholar
[10]Jacobson, N., Structure of Rings, Amer. Math. Soc. Colloq. Publ., v. 37, (Providence, R. I. 1964.)Google Scholar
[11]Kruse, R., ‘Identities satisfied by a finite ring’, J. of Alg. (to appear).Google Scholar
[12]Lyndon, R. C., ‘Two notes on nilpotent groups’, Proc. Amer. Math. Soc. 3 (1952), 579–583.CrossRefGoogle Scholar
[13]Neumann, B. H., ‘Identical relations in groups I’, Math. Ann. 114 (1937), 506–525.CrossRefGoogle Scholar
[14]Neumann, Hanna, Varieties of Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 37, (Springer Verlag, 1967.)CrossRefGoogle Scholar
[15]Werner, H. and Wille, R., ‘Charakterisierungen der primitiven Klassen arithmetischner Ringe’, Math. Z. 115 (1970), 197–200.CrossRefGoogle Scholar