Certain algebraic identities on prime rings with involution

A Pair of Derivations on Prime Rings with Antiautomorphism

“…Notably, when we set Ξ 1 = Ξ 2 and Λ = * (indicating that an antiautomorphism Λ is an involution * ) in our findings, we recover the results presented by Nejjar et al [27]. Similarly, setting Λ = * in our results yields the results of Mamouni et al [33]. Furthermore, we have demonstrated the Λ-version of Posner's second theorem [17], as seen in Theorem 1 (i).…”

“…In this article, we have successfully extended the results previously established by Nejjar et al [27] and Mamouni et al [33]. Notably, when we set Ξ 1 = Ξ 2 and Λ = * (indicating that an antiautomorphism Λ is an involution * ) in our findings, we recover the results presented by Nejjar et al [27].…”

“…In their notable work, Mamouni et al [33] (2021) made a significant contribution to the study of prime rings. They established a compelling result that holds true for prime rings denoted as R. According to their results, if such a prime ring of char(R) = 2 with an involution * of the second-kind possesses two derivations, denoted as µ, ω, that satisfies any of the following conditions:…”

On derivations over trivial extensions

“…As well as, we call the inner derivation on R into M induced by m ∈ M the map d : R → M defined by d(r) = [m, r] = m.r − r.m for every r ∈ R. In this case, if d(r) = 0 for every r ∈ M , then we say that m is a centralizer of R in M , and we write m ∈ Z M (R), where Z M (R) := {m ∈ M | r.m − m.r = 0, ∀r ∈ R}. Considerable attention has been given to the study of additive mappings and their impact on the overall structure of a ring in recent decades, including derivations, homomorphisms, and related maps (see references [1], [7], [8], [9], [10], [11]).…”

Some results on derivations and generalized derivations in rings

“…Moreover, many of obtained results extend other ones proven previously just for the action of the considered mapping on the entire ring. In this direction, the recent literature contains numerous results on commutativity in prime and semi-prime rings admitting suitably constrained derivations and generalized derivations, and several authors have improved these results by considering rings with involution (for example, see [10]). In the present paper we continue this line of investigation and study the structure of a prime ring admitting a derivations and generalized derivations satisfying more specific algebraic identities.…”