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

Abstract

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 Bra\v{s}er 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.