Definition:Graded Ring

Definition
Let $R$ be a commutative ring with unity.

Let $M$ be a monoid.

Then $R$ is an $M$-graded ring if it has an $M$-gradation.

The ring $R$ is $\N$-graded if it has a decomposition as a direct sum of abelian groups:


 * $\displaystyle R = \bigoplus_{n \in \N} R_n$

such that:
 * $\forall x \in R_n, y \in R_m: x y \in R_{m+n}$

Similarly, a ring $R$ is $\Z$-graded if there is a decomposition:


 * $\displaystyle R = \bigoplus_{n \in \Z} R_n$

such that:
 * $\forall x \in R_n, y \in R_m: x y \in R_{m+n}$