On commutativity of rings with generalized derivations

Diferential identities on Jordan ideals of rings with involution

“…Similarly, application of Theorem 3.3 yields the following result which improves ( [9], Theorem 3.8) and ([8], Theorem 2.4).…”

“…Motivated by the above results, in this section we explore the commutativity of a * -prime ring R in which the generalized derivation F satisfies similar identities on a * -Jordan ideal. We shall conclude this section with an application of our results which extend results of [8] and [9] to Jordan ideals with the additional assumption that the ring R be 2-torsion free.…”

“…Using similar arguments as used in the proof of Theorem 2.5, application of Theorem 3.1 yields the following result which extends ( [9], Theorem 3.7) and ( [8], Theorem 2.3) to Jordan ideals in the case of a 2-torsion free ring.…”

“…As an application of Theorem 2.4, the following theorem extends ( [9], Theorem 3.3) and ( [8], Theorem 2.1 ) to Jordan ideals. Moreover, if we set J = J × J, then J is a * ex -Jordan ideal of R and F [x, y] = [x, y] for all x, y ∈ J.…”

“…Reasoning as in the proof of Theorem 2.5, where F(x, y) = (F (x), −y), and using Theorem 2.6 we extend ( [9], Theorem 3.4) and ( [8], Theorem 2.2 ) to Jordan ideals as follows.…”