graded ring

Let $S$ be a groupoid (semigroup,group) and let $R$ be a ring (not necessarily with unity) which can be expressed as a $R={\oplus }_{s\in S}{R}_s$ 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 $R={\oplus }_{s\in S}{R}_s$ as an $S$-grading of$R$ and the subgroups $R_s$ as the$s$-components of $R$. If we have the strongercondition 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