By Froberg R.
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Additional info for An introduction to Grobner bases
If we apply an arbitrary homomorphism α : G → H to a G-grading Γ, then some components of Γ may coalesce in α Γ. 24. Let Γ and Γ be two gradings on V with supports S and T , respectively. We will say that Γ is a reﬁnement of Γ , or that Γ is a coarsening of Γ, and write Γ ≤ Γ, if for any s ∈ S there exists t ∈ T such that Vs ⊂ Vt . If, for some s ∈ S, this inclusion is strict, then we will speak of a proper reﬁnement or coarsening. Clearly, ≤ is a partial order on the set of all gradings on V (if we regard all relabelings as one grading).
R is said to be graded simple if R2 = 0 and the only graded two-sided ideals of R are 0 and R. A graded left R-module V is said to be graded simple or graded irreducible if RV = 0 and the only graded submodules of V are 0 and V . We emphasize that a graded simple algebra is not necessarily simple. But if an algebra is simple then it will be graded simple for any grading it might be equipped with. The same remarks apply to modules over a G-graded algebra R. In fact, it may or may not be possible to equip a given simple R-module with a G-grading that would make it a graded R-module.
Let G be a group and let R be a G-graded algebra. If R is graded simple and satisﬁes the descending chain condition on graded left ideals, then there exists a G-graded algebra D and a graded right D-module V such that D is a graded division algebra, V is ﬁnite-dimensional over D, and R is isomorphic to EndD (V ) as a G-graded algebra. Proof. Let V be a minimal graded left ideal of R. Since R is graded simple, we have either RV = 0 or R acts faithfully on V . If RV = 0, then V + V R is a nonzero graded ideal, hence R = V + V R and R2 = 0 — a contradiction.