(α, β)-Derivation on Matrix Ring Mn(R) - Dania Azzahra; Fitriani Fitriani; Ahmad Faisol (Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi, 2026)
In ring theory, a derivation is an additive mapping d:R→R satisfying Leibniz’s rule. A well-known generalization of this notion is the (α,β)-derivation, defined with respect to two ring endomorphisms α and β. In this paper, we study (α,β)-derivations on the matrix ring Mₙ(R) and several of its subrings, including scalar matrices, diagonal matrices, and upper and lower triangular matrix rings. It is shown that an (α,β)-derivation on the base ring RRR induces an (α′,β′)-derivation on these matrix subrings via entrywise extension, preserving their structural properties. Furthermore, we examine certain properties of (α,β)-derivations on the direct product ring R×R. In particular, we show that the sum of two (α,β)-derivations does not necessarily form an (α,β)-derivation, which is demonstrated through a counterexample.