The difference between the two definitions is clear in Wikipedia and nlab regarding the definition of a graduated algebra. How to explain this nuance?
From Wikipedia:
A differential ring is a ring R equipped with one or more derivations, that are homomorphisms of additive groups $\partial :R\to R\,$
such that each derivation $\partial$ satisfies the Leibniz product rule $$ \boxed{ \partial (r_{1}r_{2})=(\partial r_{1})r_{2}+r_{1}(\partial r_{2}),} $$
for every $r_{1},r_{2}\in R$ .
From nlab:
A differential algebra is an associative algebra $A$ equipped with a derivation $d \colon A \to A$, typically required to satisfy $$ \boxed{d \circ d = 0} $$.
Question :Are these two definitions equivalent or is there some semblance of generalization?
Thanks in advance for your help and support.
