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404 G. Garkusha and M. PrestLemma 5.1. If E 2 Inj A then P.E/is a homogeneous prime ideal. If the module E hasthe form E P .n/ for some prime homogeneous ideal P and integer n, then P D P.E/.Proof. The proof is similar to that of [14, 9.2].It follows from the preceding lemma that the mapP A 7! E P 2 Inj Afrom the set of homogeneous prime ideals to Inj A is injective.A tensor torsion class in Gr A is a torsion class S Gr A such that for any X 2 Sand any Y 2 Gr A the tensor product X ˝ Y is in S.Lemma 5.2. Let A be a graded ring. Then a torsion class S is a tensor torsion classof Gr A if and only if it is closed under shifts of objects, i.e. X 2 S implies X.n/ 2 Sfor any n 2 Z.Proof. Suppose that S is a tensor torsion class of Gr A. Then it is closed under shiftsof objects, because X.n/ Š X ˝ A.n/.Assume the converse. Let X 2 S and Y 2 Gr A. Then there is a surjectionLi2I A.i/f Y . It follows that 1X ˝ f W L i2IX.i/ ! X ˝ Y is a surjection.Since each X.i/ belongs to S then so does X ˝ Y .Lemma 5.3. The mapS 7! F.S/ Dfa A j A=a 2 Sgestablishes a bijection between the tensor torsion classes in Gr A and the sets F ofhomogeneous ideals satisfying the following axioms:T1: A 2 F;T2: if a 2 F and if a is a homogeneous element of A then it holds that .a W a/ Dfx 2 A j xa 2 ag 2F;T3: if a and b are homogeneous ideals of A such that a 2 F and .b W a/ 2 F forevery homogeneous element a 2 a then b 2 F.We shall refer to such filters as t-filters. Moreover, S is of finite type if and only if F.S/has a basis of finitely generated ideals, that is every ideal in F.S/ contains a finitelygenerated ideal belonging to F.S/. In this case F.S/ will be referred to as a t-filterof finite type.Proof. It is enough to observe that there is a bijection between the Gabriel filters onthe family fA.n/g n2Z of generators closed under the shift functor (i.e., if a belongs tothe Gabriel filter then so does a.n/ for all n 2 Z) and the t-filters.Proposition 5.4. The following statements are true:1. Let F be a t-filter. If I;J belong to F, then IJ 2 F.