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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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 $ {g_1}, \ldots ,{g_n}$ and one relation $ r = 1$ is free if and only if the relator r is primitive in the free group on $ {g_1}, \ldots ,{g_n}$. 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 $ u({x_1}, \ldots ,{x_n}) = v({x_1}, \ldots ,{x_n})$ is called balanced if each of u and v has the same length and number of occurrences of each $ {x_i}$. 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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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