Some Results Concerning Multiplicative (Generalized)-Derivations and Multiplicative Left Centralizers.

In ring theory, many significant studies have raised connections between the derivations and the structures of rings. Derivations of rings developed gradually more than half a century ago. In particular, the generalizations of the derivation concept play an important role in the calculation of the eigenvalues of matrices, which is important in mathematics and other sciences, business, engineering, and quantum physics. The main goal of this article is to introduce identities in prime and semiprime rings concerning left multiplicative centralizers and multiplicative (generalized)-derivations that have descriptions of these mappings. Some properties of the proposed identities are proven, and the relationships between these identities in terms of the notions of the multiplicative (generalized)-derivation (MG-D) and the multiplicative left centralizer (MLC) for an associative ring S are studied. If the condition ζ(κ1κ2)±[τ (κ1), κ2]±κ1κ2 = 0 is held for all κ1 and κ2 in an ideal J (6= 0) of S, then either ζ is an MLC on S or S is commutative. Furthermore, examples are given to show that semiprimeness and primeness are irreplaceable conditions. [ABSTRACT FROM AUTHOR]