A Pair of Generalized (α, α)-Derivations With Identities Related to Prime Ideals

Let A be an arbitrary ring, α an automorphism of A, I a nonzero ideal of A, and ϒ a prime ideal of A satisfying the condition ϒ⊊αI. This research investigates the interplay between two generalized α,α-derivations, Ω and G (associated with α,α-derivations f and h, respectively), and the resulting characteristics of the quotient ring A/ϒ. A key aspect of this work is that no primality or semiprimality conditions are imposed on the ring A. The analysis proceeds by examining specific differential identities satisfied by these derivations on the ideal I in relation to the prime ideal ϒ. Furthermore, this article discusses implications derived from the main theorems and presents nontrivial examples to underscore the necessity of the primeness hypothesis for ϒ in our results.