Results on Various Derivations and Posner's Theorem in Prime Ideals of Rings

Let R be a ring and P be a prime ideal of R. In this work, we study the structure of the quotient ring R/P in a new and more general way by discussing various algebraic identities on appropriate subsets of R involving multiplicative (generalized)-(alpha, Q)-derivations, multiplicative generalized (alpha, Q)-derivations, mul-tiplicative generalized derivations and generalized derivations. In addition, we give examples exhibiting the crucially of the hypothesis of our results.