support (graded ring)

Let $S$ be a groupoid (semigroup, group) and suppose $R={\oplus }_{s\in S}{R}_s$ is an $S$- graded ring.

We define the support of $r$ to be the set $\mathrm{supp}(r)=\{s\in S\mid r_s\ne 0\}$. We can extend this definition to $\mathrm{supp}(R)=\bigcup \mathrm{supp}{R}_s\ne 0\}$. If $\mathrm{supp}(R)$ is a finite set then we say that the ring $R$ has finitesupport.