Research

In this manuscript, we study homogeneous derivations that preserve the grading of a graded ring, which we call degree-preserving derivations (dp-derivations). We introduce the basic notions attached to dp-derivations and investigate their main properties, including dp-graded ideals, dp-closures, and dp-preimages. We then analyze structural aspects of dp-gr-simple rings and describe their behavior in both characteristic zero and positive characteristic. Furthermore, we construct dp-skew polynomial rings and show that important properties such as the gr-Noetherian condition and gr-simplicity are preserved under these extensions. Finally, we extend the theory to graded modules and develop an appropriate categorical framework for dp-graded structures.

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.

We establish results related to homogeneous derivations, a concept introduced by Kanunnikov (2018). First, we prove the existence of a non-trivial family of derivations that are not homogeneous on graded rings. Furthermore, based on homogeneous derivations, we extend certain existing significant results in the context of prime (resp. semiprime) rings to gr-prime (resp. gr-semiprime) rings, such as Posner's and Herstein's theorems.