Definition:Graded Ring

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

We say $R$ 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 if $x \in R_n$ and $y \in R_m$ then $xy \in R_{m+n}$.

Similarly, we call $R$ $\Z$-graded if there is a decomposition


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

such that if $x \in R_n$ and $y \in R_m$ then $xy \in R_{m+n}$.