Research

The concept of homogeneous derivations was introduced by Kanunnikov (2018). In this paper, we establish interesting results related to homogeneous derivations. 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.

In this paper, we introduce a novel concept in graded rings called generalized homogeneous derivations, which serve as a natural generalization of the homogeneous derivations introduced by Kanunnikov. We provide an example demonstrating that the class of homogeneous derivations is strictly contained within the class of generalized homogeneous derivations. Furthermore, we extend several significant results, originally established for prime rings, to the context of gr-prime rings.

The purpose of this paper is to study the structure of gr-prime and gr-semiprime rings through functional identities involving generalized homogeneous derivations acting on graded ideals. These mappings lead to conditions ensuring either commutativity of the ring or the existence of nonzero central graded ideals.

In this paper, we explore functional identities with central values in gr-prime rings involving pairs of homogeneous derivations. We establish commu-tativity conditions that extend classical results from prime rings to the graded setting. In particular, we show that under certain conditions for homogeneous derivations, the ring must be commutative. Furthermore, we demonstrate that these results cannot be extended to gr-semiprime rings.