graded ringGradedRing2001-10-15 18:47:262007-09-07 21:33:53Definitiongroupoid graded ringsemigroup graded ringgroup graded ringhomogeneous elementstrongly gradedalgebra ring groupoid homogeneous\usepackage{amssymb}
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\newcommand{\supp}{\,{\rm supp}\,}Let $S$ be a groupoid (semigroup,group) and let $R$ be a ring (not necessarily with unity) which can be expressed as a \PMlinkescapetext{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 {\em 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 {\em
strongly} graded by
$S$.
Any element $r_{s}$ in $R_{s}$ (where $s\in S$) is said to be {\em 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 $r_{s} \in R_{s}$.
%%We define the {\em
%%support} of $r$ to be the set $\supp(r) = \{ s \in S \st
%%r_{s} \neq 0 \}$. 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 {\em finite
%%support}.
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\\