Annals of Communications in Mathematics

AbstractBCK/BCI-algebra is a class of logical algebras that was defined by K. Iseki and S. Tanaka. BCK-algebras have a lot of generalizations. One of them is d-algebras. Near set theory which is a generalization of rough set theory. This theory is based on the determination of universal sets according to the available information of the objects. Based on the image analysis, the near set theory was created. ¨Ozt¨urk applied the notion of near sets defined by J. F. Peters to the theory of d-algebras. In this paper we introduce upper-nearness d-ideal, upper-near (upper-nearness) d#- ideal, upper-near (upper-nearness) d∗-ideal. We explored what conditions we should put on the ideal for quotient nearness d-algebra to become an nearness d-algebra again. More- over, we introduce quotient nearness d-algebras with the help of upper-nearness d∗-ideals of nearness d-algebras. Finally, we present a theorem involving the canonical homomor- phism and the structure of the kernel for nearness d-algebras. Thus, we aim to make preliminary preparations for proving isomorphism theorems for nearness d-algebras.

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.