Let R denote a ring. Let A be a commutative subset containing all elements of R with square 0; let E be the set of idempotents of R; let \(q>1\) be an integer. The first theorem asserts that R must be commutative if it has the following two properties: (i) if x,y\(\in R\) and x-y\(\in A\), then \(x^ q=y^ q\) or x and y both centralize A; (ii) \(R=.\) For the second theorem, let B be a subset of the set N of nilpotent elements, such that B is commutative or B is a proper subset of N forming a subgroup of \((R,+)\). The theorem asserts that R is commutative if for each x,y\(\in R\setminus B\) there exists p(t)\(\in Z[T]\) for which \((xy)^ 2p(xy)=yx\). There are several corollaries.