The following theorem is proved: Suppose R is a ring with identity which satisfies the identities xkyk = ykxk and xlyl = ylxl, where k and l are positive relatively prime integers. Then R is commutative. This theorem also holds for a group G. Furthermore, examples are given which show that neither R nor G need be commutative if either of the above identities is dropped. The proof of the commutativity of R uses the fact that G is commutative, where G is taken to be the group R* of units in R.