graded ring (original) (raw)

Let S be a groupoid (semigroup,group) and let R be a ring (not necessarily with unity) which can be expressed as a R=⊕s∈SRs of additive subgroups Rs of R with s∈S. If Rs⁢Rt⊆Rs⁢t for all s,t∈S then we say that R is groupoid graded (semigroup-graded, group-graded) ring.

We refer to R=⊕s∈SRs as an S-grading ofR and the subgroups Rs as thes-components of R. If we have the stronger condition that Rs⁢Rt=Rs⁢t for all s,t∈S, then we say that the ring R is strongly graded byS.

Any element rs in Rs (where s∈S) is said to be homogeneous of degrees. Each element r∈R can be expressed as a unique and finite sum r=∑s∈Srs of homogeneous elements rs∈Rs.

For any subset G⊆S we have RG=∑g∈GRg. Similarly rG=∑g∈Grg. If G is a subsemigroup of S thenRG is a subring of R. If G is a left (right, two-sided) ideal of Sthen RG is a left (right, two-sided) ideal of R.