Derivations and Commutativity in Factor Rings over Prime Ideals

In this article, we will assume that N and K are ideals in any ring G, where N is non-zero and K is prime, with K being a proper subset of N. Our main goal is to prove that if the characteristic of G/K is not 2, and G admits derivations f and h that satisfy specific functional identities, then the factor ring G/K is commutative. In some cases, we will show that the range of derivations f and h lies in K. In fact, the created identities connect a non-zero ideal N to a prime ideal K. Moreover, several important consequences and ramifications were derived. Furthermore, we will construct a counterexample to demonstrate that the conclusions of our theorems do not hold without the primeness assumption of the ideal K.