Let j, k be relatively prime positive integers. It is proved that either a group or a ring with 1 must be commutative if it satisfies the identities \(x^ jy^ j=y^ jx^ j\) and \(x^ ky^ k=y^ kx^ k\). Examples are provided to show that one of these identities for \(k>1\) is not sufficient for commutativity. For rings, certain extensions to cases where j and k vary with x and y have been obtained by the reviewer [Math. Jap. 24, 473-478 (1979; Zbl 0427.16024)].