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Search: MSC category 20N05 ( Loops, quasigroups [See also 05Bxx] )

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1. CMB 2009 (vol 52 pp. 245)

Goodaire, Edgar G.; Milies, César Polcino
Involutions of RA Loops
Let $L$ be an RA loop, that is, a loop whose loop ring over any coefficient ring $R$ is an alternative, but not associative, ring. Let $\ell\mapsto\ell^\theta$ denote an involution on $L$ and extend it linearly to the loop ring $RL$. An element $\alpha\in RL$ is \emph{symmetric} if $\alpha^\theta=\alpha$ and \emph{skew-symmetric} if $\alpha^\theta=-\alpha$. In this paper, we show that there exists an involution making the symmetric elements of $RL$ commute if and only if the characteristic of $R$ is $2$ or $\theta$ is the canonical involution on $L$, and an involution making the skew-symmetric elements of $RL$ commute if and only if the characteristic of $R$ is $2$ or $4$.

Categories:20N05, 17D05

2. CMB 2001 (vol 44 pp. 27)

Goodaire, Edgar G.; Milies, César Polcino
Normal Subloops in the Integral Loop Ring of an $\RA$ Loop
We show that an $\RA$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\RA$ loop can be normal in its unit loop. Over fields, this can never happen.

Categories:20N05, 17D05, 16S34, 16U60

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