Primitive elements and one relation algebras

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.