Some commutativity criteria for prime rings with involution involving symmetric and skew symmetric elements

Abstract

UDC 512.5 We study the Posner second theorem [Proc. Amer. Math. Soc., <strong>8</strong>, 1093–1100 (1957)] and strong com\-mu\-ta\-tivity preserving problem for symmetric and skew symmetric elements involving generalized derivations on prime rings with involution. The obtained results cover numerous known theorems. We also provide examples showing that the obtained results hold neither in the case of involution of the first kind, nor in the case where the ring is not prime.