Pair of multiplicative derivations on quotient rings with prime ideals

Let η be a prime ideal of an arbitrary ring ζ’s. We study the quotient ring <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mi>ζ</m:mi> <m:mi>η</m:mi> </m:mfrac> </m:math> {ζ/η} without assuming the primeness of ζ involving prime ideals under the action of multiplicative b -generalized derivations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">ℱ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>ζ</m:mi> <m:mo>→</m:mo> <m:msub> <m:mi mathvariant="script">𝒬</m:mi> <m:mrow> <m:mi>m</m:mi> <m:mo>⁢</m:mo> <m:mi>r</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mrow> </m:math> {F:ζₘᵣ} satisfying central algebraic identities. We also generalize Herstein’s result [I. N. Herstein, A note on derivations, Canad. Math. Bull. 21 1978, 3, 369–370, Theorem 2] in our Theorem 2.7.