On commutativity theorems for rings (Q5917412)

The author presents three commutativity theorems for rings. There are no rings satisfying the hypotheses of the first, and the second is trivial. The third, which asserts that a ring with 1 is commutative if it satisfies the identity \((x+y)^2=x^2+y^2\) and another extraneous hypothesis, is not new. In fact, \textit{C.-T. Yen} proved that if \(m\equiv 2\pmod 4\), any unital ring satisfying the identity \((x+y)^m=x^m+y^m\) is commutative [Tamkang J. Math. 21, No. 2, 123-130 (1990; Zbl 0726.16025)].