Characterizing variants of derivations for generalized quaternion ring

In this paper, we attempt to discuss the algebras of derivations, Lie derivations, generalized derivations and generalized Lie derivations of generalized quaternion ring [Formula: see text] where [Formula: see text] is a unital ring with a condition [Formula: see text]. We in principal give the results on the decomposition of above mentioned maps of [Formula: see text] in the terms of above mentioned maps of [Formula: see text] and inner derivation of [Formula: see text]. The proofs of the decomposition of Lie derivation of [Formula: see text] in terms of Lie derivation and Jordan derivation of [Formula: see text] and an inner derivation of [Formula: see text] and decomposition of generalized Lie derivation of [Formula: see text] in terms of a Lie derivation of [Formula: see text] and a left multiplier are given. Moreover, we also compute the matrix representation, admitted by the generalized Lie triple derivations of [Formula: see text] with respect to the standard basis.