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On the Lower Central Factors of a Free Associative Ring

Published online by Cambridge University Press:  20 November 2018

Robert Tyler
Robert Tyler*
Affiliation:
Susquehanna University, Selinsgrove, Pennsylvania
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Let R be a free associative ring with identity freely generated by r1, r2,. .. , rk. In analogy to group theory the lower central series for R is defined inductively By

γo = R and γn = [γn-1, R],

where γn is the ideal generated by the indicated ring commutators. Using P. Hall's collection process [2; 1, Chapter 11] γn/γn+1 will be shown to be free as a Z-module and as an R/R''-module for each non-negative integer n. In each case a basis will be exhibited.

Definition 1. Commutators of order zero are the free generators of R. A commutator, c, of order n (denoted by o(c) = n) is of the form [x, y], where x and y are commutators and o(x) + o(y) = n — 1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 2 , 01 April 1975 , pp. 434 - 438
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Hall, M., Jr., The theory of groups (The Macmillan Company, New York, 1959).Google Scholar
2. Hall, P., A contribution to the theory of groups of prime power order, Proc. London Math. Soc. 36 (1933), 29–95.Google Scholar
3. Jennings, S., Central chains of ideals in an associative ring, Duke Math. J. 9 (1942), 341–355.Google Scholar
4. Magnus, W., Karrass, A., and Solitar, D., Combinatorial group theory (Interscience, New York, 1966).Google Scholar