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

We define the support of $r$ to be the set $\supp(r) = \{ s \in S \mid r_{s} \neq 0 \}$ We can extend this definition to $\supp(R) = \bigcup \supp%%(r) = \{ s \in S \st R_{s} \neq 0 \}$ If $\supp(R)$ is a finite set then we say that the ring $R$ has finite support.