In the paper under review, a ring \(R\) is called a \(B_2\)-ring if for each 2-subset \(A\) of \(R\), \(| A^2|\leq 3\) -- that is, for each pair \(a,b\) of distinct elements of \(R\), the set \(\{a^2,b^2,ab,ba\}\) has at most 3 elements. Clearly, every commutative ring is a \(B_2\)-ring; and it is proved in [\textit{H. E. Bell} and \textit{A. A. Klein}, Arch. Math. 51, No. 6, 500-504 (1988; Zbl 0665.16021)] that every \(B_2\)-ring with 1 is commutative. The author of the paper under review gives a new proof of the previous result involving nothing more than consideration of various 2-subsets; but the proof in the previous reference is somewhat deeper. The author in the process has arrived at a new commutativity condition, called COPE property (commutativity or power-equality property). This property is that for each \(a,b\in R\), either \(ab=ba\) or there exists an integer \(n=n(a,b)>1\) for which \(a^n=b^n\). The special case in which there is a single \(n>1\) such that \(a^n=b^n\) for all noncommuting pairs \(a,b\) is called the strong COPE property. The main results are: Theorem 4. If \(R\) is any ring with COPE property, then the set of nilpotent elements of \(R\), \(N\) is an ideal and \(R/N\) is commutative. Theorem 5. Let \(R\) be a ring with 1. If \(R\) has the COPE property, then \(R\) is commutative. Theorem 6. If \(R\) is a semiprime ring with the strong COPE property, then \(R\) is commutative.