Let $S$ be a groupoid (semigroup,group) and let $R$ be a ring (not necessarily with unity) which can be expressed as a direct sum $R = {\bigoplus}_{s \in S} R_{s}$ of additive subgroups $R_{s}$ of $R$ with $s \in S$ If $R_{s} R_{t} \subseteq R_{st}$ for all $s,t \in S$ then we say that $R$ is groupoid graded (semigroup-graded, group-graded) ring.

We refer to $R = \bigoplus_{s\in S} R_{s}$ as an $S$ grading of $R$ and the subgroups $R_{s}$ as the $s$ components of $R$ If we have the stronger condition that $R_{s}R_{t} = R_{st}$ for all $s,t \in S$ then we say that the ring $R$ is strongly graded by $S$

Any element $r_{s}$ in $R_{s}$ (where $s\in S$ is said to be homogeneous of degree $s$ . Each element $r \in R$ can be expressed as a unique and finite sum $r = \sum_{s \in S} r_{s}$ of homogeneous elements $r_{s} \in R_{s}$

For any subset $G \subseteq S$ we have $R_{G} = \sum_{g \in G} R_{g}$ Similarly $r_{G} = \sum_{g \in G} r_{g}$ If $G$ is a subsemigroup of $S$ then $R_{G}$ is a subring of $R$ If $G$ is a left (right, two-sided) ideal of $S$ then $R_{G}$ is a left (right, two-sided) ideal of $R$

Some examples of graded rings include:
Polynomial rings
Ring of symmetric functions
Generalised matrix rings
Morita contexts
Ring of Hirota derivatives
group rings
filtered algebras