Definition:N-Graded Ring

Definition
Let $\N$ be the set of natural numbers.

An $\N$-graded ring or positively $\Z$-graded ring is a graded ring of type the monoid of natural numbers.

That is, it is a pair $(R, f)$ where $R$ is a ring and $f$ is a sequence $(R_n)_{n \in \N}$ of subgroups of the additive group of $R$, of which it is the internal direct sum, and such that:
 * $\forall x \in R_n, y \in R_m: x y \in R_{m+n}$

Also see

 * Definition:Z-Graded Ring