Generalized ( τ , σ )- L -Derivations in Rings

Let τ and σ : X ⟶ X be automorphisms of an arbitrary associative ring X , and let L be a prime ideal of X . The main objective of this article is to combine the notions of generalized L -derivations and ( τ , σ ) - L -derivations by introducing and analyzing a novel additive mapping Π : X → X called a generalized ( τ , σ ) - L -derivation associated with a ( τ , σ ) - L -derivation π . Later, we will examine the algebraic properties of a factor ring X / L under the influence of certain algebraic expressions containing this generalized ( τ , σ ) - L -derivation and lying in a prime ideal L . Through our main findings, we establish certain results under different conditions. It also provides various illustrative examples to show that our primeness hypotheses in various theorems are not exaggerated.