Graded Rings and Modules

1 Definitions Definition 1. A graded ring is a ring S together with a set of subgroups Sd, d ≥ 0 such that S = ⊕ d≥0 Sd as an abelian group, and st ∈ Sd+e for all s ∈ Sd, t ∈ Se. One can prove that 1 ∈ S0 and if S is a domain then any unit of S also belongs to S0. A homogenous ideal of S is an ideal a with the property that for any f ∈ a we also have fd ∈ a for all d ≥ 0. A morphism of graded rings is a morphism of rings which preserves degree. Proposition 1. Let S be a graded ring, and T ⊆ S a multiplicatively closed set. A homogenous ideal maximal among the homogenous ideals not meeting T is prime. Proof. Let T be multiplicatively closed and suppose the set Z of all homogenous ideals with a ∩ T = ∅ is nonempty. Let p be maximal in this set with respect to inclusion. It suffices to show that if a, b ∈ S are homogenous with ab ∈ p then a ∈ p or b ∈ p. If neither belong to p then (a) + p, (b) + p are homogenous ideals properly containing p, and thus both must intersect T . Say t = am+p, t′ = bn+q with t, t′ ∈ T and p, q ∈ p. Then tt′ ∈ T , but tt′ = ambn+amq+pbn+pq ∈ p since ab ∈ p. This contradiction shows that p is a homogenous prime ideal. Corollary 2. Let S be a graded ring, and T ⊆ S a multiplicatively closed set. A homogenous ideal a not meeting T can be expanded to a homogenous prime ideal p not meeting T . In particular any proper homogenous ideal is contained in a homogenous prime ideal, maximal with respect to inclusion in other proper homogenous ideals. Corollary 3. In a graded ring S the radical of a homogenous ideal a is the intersection of all homogenous prime ideals containing a.