Let Z denote the ring of integers. Let A denote a non-empty subset of a ring R and consider the following ring properties: (I-A) for each \(x\in R\) there exists p(X) in Z[X] for which \(x-x^ 2p(x)\in A\); (II-A) if x,y\(\in R\) and x-y\(\in A\), then either \(x^ 2=y^ 2\) or x and y both centralize A. The authors establish that each of the following is equivalent to the existence of a subset A for which R satisfies (I-A) and (II-A): (i) R is either commutative, or it is periodic and isomorphic to a subdirect product of nil rings of bounded index 2 and/or commutative local rings in which nilpotent elements satisfy the equations \(z^ 2=0=2z\); (ii) R is either commutative, or it is periodic and has the property that for all x,y\(\in R\) with x-y nilpotent, \(x^ 2=y^ 2\). These results are slight extensions of recent work of Y. Hirano, H. Tominaga, the authors, and the reviewer.