Abstract:

In this paper, we investigate the structural properties of symmetric bi-semiderivations in the setting of prime and semiprime rings. By adapting classical derivation identities through the use of associated ω-homomorphisms, we obtain several characterization results that generalize known theorems from derivations to the bi-additive context. Particular attention is given to the interplay between Jordan-type structures and standard bi-semiderivations. In this direction, we prove that if R is a prime ring with char(R) ≠ 2, then every mapping satisfying the symmetric Jordan bi-semiderivation identity is in fact a symmetric bi-semiderivation. To illustrate our results, we present explicit examples constructed from matrix rings and polynomial rings, which also highlight the necessity of the imposed algebraic conditions.