Let $S$ be a groupoid (semigroup, group) and suppose $R=\bigoplus_{s\in S}R_{s}$ is an $S$- graded ring.
We define the of $r$ to be the set $\,{\rm supp}\,(r)=\{s\in S\mid r_{s}\neq 0\}$. We can extend this definition to $\,{\rm supp}\,(R)=\bigcup\,{\rm supp}\,R_{s}\neq 0\}$. If $\,{\rm supp}\,(R)$ is a finite set then we say that the ring $R$ has finite support.