On a problem of commutativity of automorphisms (Q1384268)

Summary: We provide a partial answer to a problem proposed by M. Brešar. We prove that if \(\alpha\), \(\beta\) are automorphisms of a commutative prime ring of characteristic not equal to 2 satisfying the equation \(\alpha+\alpha^{-1}=\beta+\beta^{-1}\), then either \(\alpha=\beta\) or \(\alpha=\beta^{-1}\). As a consequence \(\alpha\) and \(\beta\) commute and in this situation the equation itself ensures the commutativity of \(\alpha\) and \(\beta\).