\begin{definition}[Definition:N-Graded Ring]
Let $\N$ be the set of natural numbers.
An '''$\N$-graded ring''' or '''positively $\Z$-graded ring''' is a graded ring of type the additive monoid of natural numbers.
That is, it is a pair $\struct {R, f}$ where:
:$R$ is a ring
:$f$ is a sequence $\sequence {R_n}_{n \mathop \in \N}$ of subgroups of the additive group of $R$, of which it is the internal direct sum, and such that:
::$\forall x \in R_n, y \in R_m: x y \in R_{m + n}$
\end{definition}