Is this a equivalent definition for a graded ring?
Let $R$ be a ring. We say that $R$ is graded if there are subgroups $R_n, n\in \mathbb{Z}$ of $R$ such that given $x\in R$, there are $x_n \in R_n$ such that $x=\sum x_n$ where the sum is finite, i.e., there are only a finite number of non-zero $x_n$ such that this expression is unique.
The definition I am familiar is:
$R$ is graded if $R=\bigoplus R_n$ as abelian groups and $R_nR_m\subset R_{m+n}$