Moufang loops with limited commutativity and one commutator (Q1093018)

A Moufang loop has the limited commutativity property if whenever two elements x and y commute, one of x, y, xy is central. This paper represents the culmination of previous work of the authors dealing with the structure of Moufang loops whose loop rings, over an associative ring of characteristic \(\neq 2\), are alternative rings. [See \textit{E. G. Goodaire}, Publ. Math. 30, 31-38 (1983; Zbl 0537.17006)], and \textit{O. Chein} and \textit{E. G. Goodaire}, Commun. Algebra 14, 293-310 (1986; Zbl 0582.17015).] The main theorem lists six conditions equivalent to the property that a loop L should have an alternative loop ring over some ring of characteristic \(\neq 2\). One of the more curious of these is that L should be a Moufang loop with the limited commutativity property and a unique nonidentity commutator.