On the Lie derivations and generalized Lie derivations of quaternion rings

Lie triple derivation and Lie bi-derivation on quaternion rings

“…Upon componentwise addition and multiplication with respect to the given condition, and the conventions that i, j, k commute with W elementwise, H (W ) is a ring called the quaternion ring over W . [6,Lemma 2.1]). It is significant to mention that the quaternion ring Q = H (W ) turns out to be (isomorphic to) a 2 × 2 full matrix ring in some specific cases.…”

“…In 2020, Ghahramani et al [6] evaluated Lie derivations on quaternion rings and demonstrated that every Lie derivation is in standard form on acceptable quaternion rings. Again in 2021, Ghahramani et al [7] demonstrated that every Jordan derivation of W is a derivation, and that every derivation of W decomposes into the sum of an inner derivation and a derivation induced by a derivation on W .…”

On Lie Derivations, Generalized Lie Derivations and Lie Centralizers of Octonion Algebras

“…Our next task is to present characterization of Lie derivations of the algebra of Octonion. In Theorem 2.2 of [21], it is shown that if S be a 2-torsion free ring and R = H(S ) be quaternion ring, then every Lie derivation of R can be decomposed in terms of Jordan derivation and Lie derivation of S and an inner derivation of R, for every element t ∈ R. Here, we have: Proof. Since D is an additive map, we can write…”

“…Authors characterized in [19], the Lie triple derivations of algebra of tensor product of some algebra T and quaternion algebra. Ghahramani et al in [20] proved results on the characterization of generalized derivation and generalized Jordan derivation of ring of quaternion and in [21] discussed the characterization of Lie derivation and its natural generic extension of quaternion ring. This article is arranged in the following order: Section 2 contains some minor details of Octonion algebras equipped with commutator product denoted by O.…”

On the structure of quaternion rings

“…The first two authors proved in [3] that if S is a ring whose characteristic is an odd prime number, then the quaternion ring H(S) is isomorphic to the 2 × 2 matrix ring M 2 (S). Moreover, recently Ghahramani et al [8] described the form of some mappings on quaternion rings. This paper is organized as follows.…”