Primitive elements and one relation algebras

Author:
Catherine Aust

Journal:
Trans. Amer. Math. Soc. **193** (1974), 375-387

MSC:
Primary 08A15

DOI:
https://doi.org/10.1090/S0002-9947-1974-0344176-2

MathSciNet review:
0344176

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Abstract: Let *F* be a free algebra in a variety *V*. An element *p* of *F* is called *primitive* if it is contained in some free generating set for *F*. In 1936, J. H. C. Whitehead proved that a group with generators and one relation is free if and only if the relator *r* is primitive in the free group on . In tnis paper, tne question of whether there is an analogous theorem for other varieties is considered. A necessary and sufficient condition that a finitely generated, one relation algebra be free is proved for any Schreier variety of nonassociative linear algebras and for any variety defined by balanced identities. An identity is called *balanced* if each of *u* and *v* has the same length and number of occurrences of each . General sufficiency conditions that a finitely generated, one relation algebra be free are given, and all of the known results analogous to the Whitehead theorem are shown to be equivalent to a general necessary condition. Also an algebraic proof of Whitehead's theorem is outlined to suggest the line of argument for other varieties.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0344176-2

Keywords:
Primitive element,
free algebra,
one relation algebra,
elementary transformation,
Schreier variety of nonassociative linear algebras,
balanced identity,
balanced variety

Article copyright:
© Copyright 1974
American Mathematical Society