A ring R is called almost-quasi-commutative if for each x, y ∈ R there exist nonzero relatively prime integers j = j(x, y) and k = k(x, y) and a non-negative integer n = n(x, y) such that jxy = k(yx)n. We establish some general properties of such rings, study commutativity of almost-quasi-commutative R, and consider several examples.
I. N. Herstein  proved that a prime ring of characteristic not two with a nonzero derivation d satisfying d(x)d(y) = d(y)d(x) for all x, y must be commutative, and H. E. Bell and M. N. Daif  showed that a prime ring of arbitrary characteristic with nonzero
derivation d satisfying d(xy) = d(yx) for all x, y in some nonzero ideal must also be commutative. For semiprime rings, we show that an inner derivation satisfying the condition
of Bell and Daif on a nonzero ideal must be zero on that ideal, and for rings with identity, we generalize all three results
to conditions on derivations of powers and powers of derivations. For example, let R be a prime ring with identity and nonzero derivation d, and let m and n be positive integers such that, when charR is finite, m ∨ n < charR. If d(xmyn) = d(ynxm) for all x, y ∈ R, then R is commutative. If, in addition, charR≠ 2 and the identity is in the image of an ideal I under d, then d(x)md(y)n = d(y)nd(x)m for all x, y ∈ I also implies that R is commutative.
We give a characterization of all those commutative groups which admit at least one absolutely nonmeasurable homomorphism into the real line (or into the one-dimensional torus). These are exactly those commutative groups (G, +) for which the quotient group G/G0 is uncountable, where G0 denotes the torsion subgroup of G.
For a finite abelian group A, the ring End(A) is a maximal ring in the nearring M(A). In this paper we consider an analogous problem for finite semigroups, concentrating on commutative Clifford semigroups.