Zero commutativity of nilpotent elements via tripotents

We study $$\textrm{CNZ}$$ properties of rings in a more general setting by introducing the concept of t- $$\textrm{CNZ}$$ rings, defined via a ring tripotent, and investigating their properties. We prove that every $$\textrm{CNZ}$$ ring is a t- $$\textrm{CNZ}$$ ring, but the converse fails, as we demonstrate by a counterexample. Additional examples and counterexamples are provided to illustrate these results. Moreover, we examine t- $$\textrm{CNZ}$$ properties of rings relative to a ring endomorphism $$\alpha $$ . Some results on reversible rings, $$\textrm{CNZ}$$ rings and $$\alpha $$ -skew $$\textrm{CNZ}$$ rings are extended and unified (see [1, 2] and [4]).