Generalized Homogeneous Derivations on Graded Rings
Yassine Ait Mohamed
TL;DR
The paper develops a comprehensive theory of generalized homogeneous derivations on $G$-graded rings, extending Kanunnikov's homogeneous derivations to a graded setting and preserving homogeneous components via $F(xy)=F(x)y+x d(y)$. It introduces gr-generalized derivations, analyzes their algebraic and Lie structures, and establishes functorial and categorical frameworks that describe their behavior on rings and modules, including how they interact with graded ideals, quotients, and tensor products. The authors extend classical commutativity and Posner-type results from prime rings to gr-prime and gr-semiprime rings, providing existence criteria for central graded ideals and highlighting the necessity of graded hypotheses through examples. The work culminates with a graded module theory, defining gr-generalized derivations on modules and constructing the category $ ext{M}^{gh}_G$, thereby offering a coherent, categorical perspective on graded derivations with broad implications for graded algebra and representation theory.
Abstract
We introduce a notion of generalized homogeneous derivations on graded rings as a natural extension of the homogeneous derivations defined by Kanunnikov. We then define gr-generalized derivations, which preserve the degrees of homogeneous components. Several significant results originally established for prime rings are extended to the setting of gr-prime rings, and we characterize conditions under which gr-semiprime rings contain nontrivial central graded ideals. In addition, we investigate the algebraic and module-theoretic structures of these maps, establish their functorial properties, and develop categorical frameworks that describe their derivation structures in both ring and module contexts.