Generalized reverse derivations and commutativity of prime rings

Abstract

Let R be a prime ring with center Z(R) and I a nonzero right ideal of R. Suppose that R admits a generalized reverse derivation (F, d) such that d(Z(R)) = 0. In the present paper, we shall prove that if one of the following conditions holds:

Commutativity of prime rings with generalized (α,β)-reverse derivations satisfying certain identities

“…But β is an automorphism on R so we can replace y by β −1 (m) in the last equation to get [m, x] ∈ Z(R) for all m, x ∈ I. Using arguments which used in [14] Theorem 1, we get the required result.…”

“…In 2018 Ozge, Atay [10] and S. Huang [14] studied the commutativity of prime rings R admitting a generalized reverse derivation F associated with reverse derivation d satisfying several identities on an appropriate subset of R. In 2018, Merve and Aydm [8] studied some properties of (α, β)-reverse derivations on prime and semiprime rings.…”

On Multiplicative (Generalized)-Derivations and Central Valued Conditions in Prime Rings

“…During the last six decades there have been many results showing that the global structure of a ring is often tightly connected with the behaviour of additive and multiplicative mappings defined on it (see [2], [9], [10], [11], [12], [13]). In 1957, Posner [11] initiated the study of identities involving derivations that ensure commutativity.…”

“…, where 1 id is the identity map of R. Very recently, Huang [10] explored the commutativity of prime rings with specific additive mapping F that satisfy the following identities: (i)…”

“…Therefore, in this view it is the optimal case to study certain algebraic identities over one-sided ideals of rings. In the present paper, motivated by Huang [10] and Dhara and Pradhan [8], we study a number of central valued conditions with multiplicative (generalized)-derivations on one-sided ideals of prime rings and obtain the commutativity of R.…”