Let R be an associative ring and Z(R) the center of R. In this paper, our primary purpose is to investigate certain central-valued identities involving generalized derivations of associative rings containing a prime ideal. More precisely, we study how these identities act on suitable subsets of the ring and analyze their structural consequences. It is shown that, under some mild and natural assumptions, such identities either force the corresponding factor rings to be commutative or lead to a precise description of the generalized derivations involved. Our results unify and extend several known commutativity theorems concerning derivations and generalized derivations in prime and semiprime rings. Furthermore, the obtained characterizations provide deeper insight into the interaction between ring structure and central-valued conditions, and highlight the role of prime ideals in governing such behavior. As an application of our main results, we obtain an affirmative answer to a particular case of the open problems posed by M. Zerra, K. Bouchannafa, and L. Oukhtite in Georgian Math. J. 32, 175–182 (2024).
Gurninder S. Sandhu (Sun,) studied this question.
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