# 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 $R={\bigoplus}_{s\in S}R_{s}$ 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 $R$.

 Title graded ring Canonical name GradedRing Date of creation 2013-03-22 11:45:03 Last modified on 2013-03-22 11:45:03 Owner aplant (12431) Last modified by aplant (12431) Numerical id 19 Author aplant (12431) Entry type Definition Classification msc 13A02 Classification msc 16W30 Classification msc 14L15 Classification msc 14L05 Classification msc 12F10 Classification msc 11S31 Classification msc 11S15 Classification msc 11R33 Synonym S-graded ring Synonym G-graded ring Related topic HomogeneousIdeal Related topic SupportGradedRing Defines groupoid graded ring Defines semigroup graded ring Defines group graded ring Defines homogeneous element Defines strongly graded