Symmetric Units in Alternative Loop Rings

Abstract

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. For α = α in a loop ring RL, define α = α −1 and call α symmetric if α = α. We find necessary and sufficient conditions under which the symmetric units are closed under multiplication (and hence form a subloop of the loop of units in RL) when R has characteristic two and when R = Z, the ring of rational integers. CNPq., Proc. 300243/79-0(RN) of Brasil. Algebra Colloq. 2006.13:361-370. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/05/15. For personal use only. Algebra Colloq. 2006.13:361-370. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/05/15. For personal use only.

Central units in salternative loop rings

“…Replacing L by the support of µ and, if necessary, three elements which do not associate, we may assume that L is finitely generated, thus T is finite and the loop algebra QT = A i is the direct sum of unique simple algebras A i with identity elements e i which are primitive central idempotents. (This is classical if T is associative and Corollary VI.4.3 in [2] if T is an RA loop.) Writing e i ∼ e j if e j = e i θ for some inner map θ, we obtain an equivalence relation and, partitioning the e i into equivalence classes, we have QT = m h=1 R h , each R h a sum of those A j whose identities form an equivalence class.…”

“…(In group rings, α θ is denoted α * , notation which we avoid because α * has an entirely different meaning when L is an RA loop [2,§III.4].) It is easy to see that θ defines an antiautomorphism of ZL.…”

“…A good reference for the theory of RA loops (and the loop rings they determine) is [2], though it is convenient to record here a few properties of RA loops that are of particular relevance to this paper.…”

Hypercentral units in alternative loop rings

“…By now, RA loops have been completely classified and many properties of the associated alternative loop rings explored. The best source of information on these subjects is [1] to which we make frequent reference here. We record now some properties of RA loops of special interest in this paper.…”

“…An RA loop L has the LC property: elements g, h ∈ L commute if and only if one of g, h, gh is central [1,§IV.2]. In particular, this implies that the square of any element of an RA loop is central.…”