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402 G. Garkusha and M. Prestof Inj zg R and torsion classes of finite type in Mod R. However, this is not the case forgeneral commutative rings.The topology on Inj zg R can be defined as follows. Let M be the set of those modulesM which are kernels of homomorphisms between finitely presented modules; that isM D Ker.Kf ! L/ with K; L finitely presented. The sets .M / with M 2 M forma basis of open sets for Inj zg R. We claim that there is a ring R and a module M 2 Msuch that the intersection .M / \ Spec R is not open in Spec R, and hence such thatthe open subset .M / cannot correspond to any torsion class of finite type on Mod R.Such a ring has been pointed out by G. Puninski.LLet V be a commutative valuation domain with value group isomorphic to Dn2ZZ,aZ-indexed direct sum of copies of Z. The order on is defined as follows..a n / n2Z >.b n / n2Z if a i >b i for some i and a k D b k for every k.an 0 / n for some .an 0 / n with an 0 D 0 for all n N for some fixed N . (Recall thatthe valuation v on R satisfies v.r C s/ minfv.r/; v.s/g and v.rs/ D v.r/ C v.s/.)It follows that there is a prime ideal properly between I and J.R/. This gives acontradiction, as required.5 Graded rings and modulesIn this section we recall some basic facts about graded rings and modules.Definition. A(positively) graded ring is a ring A together with a direct sum decompositionA D A 0 ˚ A 1 ˚ A 2 ˚ as abelian groups, such that A i A j A iCj fori;j 0. Ahomogeneous element of A is simply an element of one of the groups A j ,and a homogeneous ideal of A is an ideal that is generated by homogeneous elements.A graded A-module is an A-module M together with a direct sum decompositionM D L j 2Z M j as abelian groups, such that A i M j M iCj for i 0; j 2 Z. Onecalls M j the j th homogeneous component of M . The elements x 2 M j are said to behomogeneous (of degree j ).