Wedderburn’s Theorem,
asserting that a finite associative division ring is necessarily commutative, has been
extended to
Theorem 1. Let R be a noncommutative Jordan ring of characteristic not 2, and
let I be an ideal in R such that R∕I is a finite division ring of characteristic p > 5
with exactly q elements. Suppose that (i) I is commutative and every associator
contained in the ideal generated by I2 vanishes, and (ii) x ≡ y (modI)
implies xq= yq or both x and y commute with all elements of I. Then R is
commutative.
The object of this paper is to extend Theorem 1 in two directions. First we
replace the assumption that R is a noncommutative Jordan ring by the weaker
assumption that R is power-associative. Next we assume that R is a flexible
power-associative ring but replace the hypothesis that every associator in
the ideal generated by I2 vanishes with the weaker assumption that I is
associative. In each case we drop the assumption that R is of characteristic not
2.
