Structure and Applications of $(p,q)$-Derivations in Algebra

In this paper, we investigate $(p,q)$-derivations in both general algebras and Banach algebras. Our main result extends the Singer-Wermer theorem to $(p,q)$-derivations, proving that their ranges are contained in the radical of the algebra. As a direct consequence, we prove that the range of any left derivation lies within the radical of the algebra even without assuming continuity. This improves a result due to Brašer and Vukman. For Banach algebras, we show that primitive ideals remain invariant under bounded $(p,q)$-derivations. Finally, we study $(p,q)$-derivations of group algebras and give an answer to the question posed.