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)-(α; β)-derivations, multiplicative generalized (α; β)-derivations, multiplicative generalized derivations and generalized derivations. In addition, we give examples exhibiting the cruciality of the hypothesis of our results.