THE LATTICE R-tors FOR PERFECT RINGS

THE LATTICE R-tors FOR PERFECT RINGS HUGO ALBERTO RINCÓN-MEJÍA We define ^'F in R-tors by r ^'F a iff the class of r-codivisible modules coincides with the class of o-codivisible modules . We prove that if R is left perfect ring (resp . semiperfect ring then every [r1 p E R-tors/-F (resp . [X]F and KIF) is a complete sublattice of R-tors . We describe the largest element in [r] as X(Rad R/t,(Rad R)) and the least element of [r] as 1(t r(RadR» . Using these results we give a necessary and sufficient condition for the central splitting of Goldman torsion theory when R is semiperfect. We prove that for a QF ring R the least element of [X]^'F is the Goldie torsion theory. This can be used to prove that for a QF ring ^'F and -T are equal, where r ^'T o iff the class of r-injective modules coincides with the class of v-injective modules . 0. Introduction Throughout this work R will denote an associative unital ring ; R-tors will denote the complete brouwerian lattice of all left hereditary torsion theories ; X (resp . 1) will denote the largest (resp . the smallest) element of R-tors . If {Ma}aEx is a family of left R-modules, then X({Ma }) will denote the largest torsion theory respect to which every Ma is torsion free . J({M« }) will denote the smallest torsion theory respect to which every Ma is torsion . We consider a torsion theory r as an ordered pair r = (T r , F r ), where T,. denotes the class of r-torsion modules, and Fr denotes the class of r-torsion free modules . Also remember that the order in R-tors is given by : r < o, iff Tr C To . Remember that a left module M is r-codivisible iff ExtR(M,K) = (0) VK E F r . Let us denote P r the class of r-codivisible modules . We define ^'F in Rtors by r ^'F a iff Pr = Po . Obviously this is an equivalence relation in R-tors . Our aim in this work is to study R-tors by looking at the equivalence classes [r] E R-tors/-F . In case R is a left perfect ring, these equivalence classes are complete sublattices of R-tors . So, in [r] there must exist a largest element (resp . a smallest element) which will be denote r* (resp . r* ) . We describe r* = x(Rad Rltr (Rad R)) (resp . r* = J(tr(Rad R))), where Rad R denotes the Jacobson radical of R. 18 H.A . RINCÓN-MEJÍA We also obtain some generalizations of some results of Bland (see 3) . We also prove that for a QF-ring R the smallest element of [X]-F (which exists, since R is left perfect) is Goldie's torsion theory . In fact, it can be proved that for a QF-ring R the equivalente relations ^'F and -T coincide, where we define T -T Q iff the class of r-injective modules coincides with the class of Q-injective modules . The partition R-tors/^'T has been studied by Raggi & Ríos (see [12] and [131) . We will denote by ST the class of all short exact sequences 0 --> K -> L -r M --> 0 in R-mod such that K E F T , where r E R-tors . We will denote PT the class of R-modules that are projective with respect to each sequence in S, . We will denote AT the proper class of short exact sequences in R-mod which make projective each element of Pr . We should observe that RP is projective with respect to each short exact sequence in ST ~? P is projective with respect to each element of A, . Remarks . 1) (Ohtake [10], Bican, Nemec, Kepka [2]) . If T = (T, F) E R-tors and 0 --> h -> P ---) M -> 0 is a short exact sequence in R-mod such that P is projective an K E T, then M E P T . 2) R-mod has enough A,-projectives (this means that VRM E Rmod 3 0 -) K --> P -> M ---) 0 E Ar with P projective with respect to A, . 3) Let RM E R-mod. Then: M E Pr <--~ M is a direct summand of a module of the form P/T, where P is projective and T E TT . We should observe that in the above remark we can replace "projective" by "free" . Definition 1 . (r-codivisible cover, Bland [3]) . An . 4 r -projective cover of RM is an exact sequence 0 --, L --~ P M -> 0, such that i)LEFr . ü) P is T-codivisible (¡ .e . .~4,-projective). iii) i(L) is small in P (i(L) « (P) . The faci of that r-codivisible cgvers are unique except for isomorphic copies is a known result (3J . We will denote by 0 -> KT(M) ---> P, (M) -r M -r 0 the T-codivisible cover of M, when it exists, and by 0 --~ K(M) --~ P(M) 3 M -> 0 the projective cover of M, when it exists . Definition 2. We define ^'F in R-tors by : Q ^'F T iff AQ = Ar (or equivalently, if P o = Pr, i.e . if the class of Q-codivisible modules coincides with ¡he class of r-codivisible covers) . The relation defined above is, obviously, an equivalente relation . Under appropiate conditions the corresponding equivalence classes [T]-F, are complete sublattices of R-tors . This is the case when R is a left perfect ring . Theorem 1. If 0 --> K,(M) -) P,(M) -> M --> 0 is a T-codivisible cover of M and if 0 ) K(M) )P(M) -~ M 0 is a projective cover of M, then ker(P(M) )PT(M)) is T-torsion. Lemma 1. Let 0 ) K ) P)M) 0 be a projective cover. Let us Suppose T ^'F v, then K E Tr K E To . Proof. Straightforward . Theorem 2 . Suppose that 0 --) K(M) -) P(M) ) M -> 0 is a projective cover. Then 0 ) K(M)/tr(K(M)) ) P(M)/t,(K(M)) M)0 (*) is a a-codivisible cover Va E [T]F . Proof. Direct from the definitions . LATTICE R-TORS 19 Note that the above theorem implies that if 0 ) K,(M) ) P(M) ) M ) 0 is a T-codivisible cover, then K,(M) E Fvj, , s. This is because K,(M) E n[TIF,, = Fv[T]o . Let us also note that the following implications hold for u, r E R-tors : T<Q~FTDF,==> A,DA,~P,CP, Remarks. For a proper class A we have: i) A = Af -4~ A is the class of all short exact sequences in R-mod 4==> PA = P{ . Also note that PI, the class of J-codivisible modules is precisely the class of all projective modules. ii).,4=A¿ S,4={0)0)M)M 0 :MER-mod}4=> R-mod = PA, the class of all projective modules. Also note Ax is the class of all splitting short exact sequences in R-mod. iii) -r E R-tors faithful ==> r E [1] : for if P is T-codivisible, then P is a direct summand of a module R(X)/T, where T is a T-torsion submodule of R(X) , which is in FT (being R in Fr, by hypothesis) . Then T = 0, and hence P is a direct summand of a free module ; Le ., P is projective . So P{ = P, and we conclude by using i) . iv) If R is a domain (e.g . Z) every X qÉ T E R-tors is faithful and hence is in KIF . So R-tors/^'F has only the two elements [X]F = {X}, and [fF = R-tors\{X} . Moreover [~] has a maximal member: X(R) = TL, Lambek's torsion theory. v) For a stable torsion theory T the following statements are equivalent : a) R tr(R) x S, where S is semisimple artinian . b) T E [X]F20 H.A . RINCÓN-MEJÍA to : c) `dN E FT , N is an injective semisimple module . Proof. a) b) (See [111), b) t=> c) follows from Theorem 3. vi) For a left semiartinian ring are equivalent a) -rG E [X] ( TG denotes Goldie's torsion theory) . b) R = TG(R) x S, where S is semisimple artinian . c) rG centrally splits . d) To is stable . Here -ro denotes Goldman's torsion theory ; Le ., the torsion theory generated by the projective semisimple modules. Proofb) c) d) (See [111) . a)~! b) follows from Remark v) . vi¡) If R is right perfect ring, then the above conditions are also equivalent e) soc p (Rad R) = 0 (See Theorem 18) . Here socp denotes the projective socle, and Rad R denotes the Jacobson radical . The following is an easy generalization of a Theorem of Bland, in our context . Theorem 3. Are equivalent for T E R-tors : i) z E [x] . ü) PT = Px = R-mod . iii) AT = class of all splitting short exact sequences . iv) VRN E FT, N is semisimple and injective . v) The ring R/t,(R) is semisimple . vi) All cyclic modules are ,,4 r -projective . (Bland in (3) shows the equivalente of ii), iv) and v), the equivalente of ¡he others follows direcily from the definitions) . Corollary 1. R is semisimple R-tors/^'F = {[l;]}(!~ 1 ^'F X) . Proof. ==> ) If R is semisimple, then dT E R-tors, R/tr(R) is semisimple ; so by v) ==> i) in Theorem 3 we get T E [X]F . Hence [1] = [X] = R-tors . ~ ) If R-tors/^'F = {[1]} . In particular 1 E [X] = [l;] . So by using i) ~? iv) in the above theorem, we get N is semisimple b'RN E F{ (but F{ = R-mod) . Then R is semisimple . From the preceeding coróllary, we obtain immediately the following result . Corollary 2 . (Bland ¡3J, Corollary 3 ./, proves the "if" par¡) . R is semisimple <===> 3T E [X], faithful . Proof: ===> ) If R is semisimple, then 1 has the required properties . If T E [X] is faithful, then we get that -r E [l;] (see Rmark iii), after Theorem 2) . Thus T E [¿] n [X] . Hence [1] = [X] . Theorem 4 . Let T be an element of R-tors . Then [ 7IF is elosed under finite meets. Proof. Let us suppose that -rl -F r2 -F r . By the observation after Theorem 2 we have that A rl C ,,4,,,\, (ri A T2 < r2 ) . Now, let us consider the diagram 0 -~ L~M 3 MIL:0 with L E FT1 ,,,, S E P rl , and remember that S is A,-projective iff S is projective with respect to each exact sequence of the form 0 ---> L ---+ M--~ N --> 0 with L E F, Let us extend the above diagram to L LATTICE R-TORS 2 1


Introduction
Throughout this work R will denote an associative unital ring; R-tors will denote the complete brouwerian lattice of all left hereditary torsion theories ; X (resp .1) will denote the largest (resp .the smallest) element of R-tors.
If {Ma}aEx is a family of left R-modules, then X({Ma }) will denote the largest torsion theory respect to which every Ma is torsion free.J({M« }) will denote the smallest torsion theory respect to which every Ma is torsion .We consider a torsion theory r as an ordered pair r = (T r , F r ), where T,. denotes the class of r-torsion modules, and Fr denotes the class of r-torsion free modules .Also remember that the order in R-tors is given by: r < o, iff Tr C To.
Remember that a left module M is r-codivisible iff ExtR(M, K) = (0) VK E F r .Let us denote P r the class of r-codivisible modules .We define ^'F in Rtors by r ^'F a iff Pr = Po .Obviously this is an equivalence relation in R-tors.Our aim in this work is to study R-tors by looking at the equivalence classes [r] E R-tors/-F .In case R is a left perfect ring, these equivalence classes are complete sublattices of R-tors.So, in [r] there must exist a largest element (resp.a smallest element) which will be denote r* (resp .r* ) .We describe r* = x(Rad Rlt r (Rad R)) (resp .r* = J(t r(Rad R))), where Rad R denotes the Jacobson radical of R.
We also obtain some generalizations of some results of Bland (see 3).We also prove that for a QF-ring R the smallest element of [X]-F (which exists, since R is left perfect) is Goldie's torsion theory.In fact, it can be proved that for a QF-ring R the equivalente relations ^'F and -T coincide, where we define T -T Q iff the class of r-injective modules coincides with the class of Q-injective modules .
We will denote by ST the class of all short exact sequences 0 --> K -> L -r M --> 0 in R-mod such that K E F T , where r E R-tors.
We will denote PT the class of R-modules that are projective with respect to each sequence in S, .
We will denote AT the proper class of short exact sequences in R-mod which make projective each element of Pr.
We should observe that RP is projective with respect to each short exact sequence in ST ~? P is projective with respect to each element of A,.
3) Let RM E R-mod.Then: M E Pr <--~M is a direct summand of a module of the form P/T, where P is projective and T E TT .
The relation defined above is, obviously, an equivalente relation .Under appropiate conditions the corresponding equivalence classes [T]-F, are complete sublattices of R-tors .This is the case when R is a left perfect ring.
Proof.Direct from the definitions.
Note that the above theorem implies that if 0 ) K,(M) Let us also note that the following implications hold for u, r E R-tors : T<Q~FTDF,==> A,DA,~P,CP, Remarks.For a proper class A we have: i) A = Af -4~A is the class of all short exact sequences in R-mod 4==> PA = P{ .
Also note that PI, the class of J-codivisible modules is precisely the class of all projective modules.ii).,4=A¿S,4={0)0)M)M 0 :MER-mod}4=> R-mod = PA, the class of all projective modules.Also note Ax is the class of all splitting short exact sequences in R-mod.iii) -r E R-tors faithful ==> r E [1]: for if P is T-codivisible, then P is a direct summand of a module R(X)/T, where T is a T-torsion submodule of R(X) , which is in FT (being R in Fr, by hypothesis) .Then T = 0, and hence P is a direct summand of a free module ; Le., P is projective.So P{ = P, and we conclude by using i).
iv) If R is a domain (e.g.Z) every X qÉ T E R-tors is faithful and hence The following is an easy generalization of a Theorem of Bland, in our context .
iii) AT = class of all splitting short exact sequences .iv) VRN E FT, N is semisimple and injective .v) The ring R/t,(R) is semisimple .vi) All cyclic modules are ,,4 r -projective .
(Bland in (3) shows the equivalente of ii), iv) and v), the equivalente of ¡he others follows direcily from the definitions) .
Then R is semisimple .
From the preceeding coróllary, we obtain immediately the following result .with L E FT1,,,, S E P rl , and remember that S is A,-projective iff S is projective with respect to each exact sequence of the form 0 where 7r is the natural epimorphism .Now MIt2(L) E Fr2 ; so 0 ---" ker7r -r MIt2(L) nr MIL ---3 0 E Are = Al, .Inasmuch as S is in P, = Pr2 , we have that 30: S --> MIt2(L), such that 7r o /3 = a.Now let us observe that But in the other hand, tl (t 2 (L)) C_ L E F Tl .T2 ; hence t l (t 2 (L)) = 0.So t2 (L) E F rl , which implies that 0 --> t2(L) --r M -) Mlt2(L) ---> 0 belongs to Ar1 .Hence 3 -y : S --> M such that p o -y = /i; so the following diagram is commutative : But then y o p = 7r o p o y = 7r o fl = a .Hence S E P T, A,,, and then Prl C_ Pr,nr2 , and from this we get Ar,Ar2 C .,4r (see the observation after Theorem 2).
If the ring R is left perfect we can prove much more.By the preceeding theorem, we know that if R is a left perfect ring, then [T] is closed under taking arbitrary joins and meets.Consequently, in [T] must exist a largest and a smallest element, which will be denoted T* and T* , respectively.The following theorem gives us a useful description of each of them.For the particular cases when T E {1;, x} and when the ring R is left perfect, we give descriptions of r* and r* by using the Jacobson radical of R, which we will extend to arbitrary torsion theories and for semiperfect rings.Theorem 9.For left perfect R we have that i) e* = x(J(R)) ii) x* = «J(R)), where J(R) denotes the Jacobson radical of R.
We give now more "concrete" descriptions of r* and r*, in case R is left perfect .
Theorem 10 is extended in [14] to the case of local rings .In that situation each [r] E Rtors/^'F is closed under taking joins and meets and moreover the biggest element in [7-1, ,r* is given by r* = X(,7(R)/t,(J(R))) and also r* = J(tT(J(R))) .
However, a ring may have the property of having each [Q]F closed under arbitrary joins and meets without being semiperfect .Moreover, the elements Q* and o* are not given by X(,7(R)/t,(J(R))) and by j(t,(J(R))), in general .As we see in the following examples .
Examples .In view of Remark 3 before Definition 1, is easy to see that if R is a domain, then R-tors admits the following partition : It is clear that each equivalence class in Rtors/^'F admits a largest and a least element .
In particular this is the situation for Z, the ring of integers, which is not a perfect ring.
Moreover, let us note that for Z, in spite of the fact that each element in R-tors/^'F has a largest and a least element, they are not given as in Theorem 10 .Explicity, ,%(Z) = 0, but we have that [X] = {X}, and so X* = X = X*.Nevertheless X* :~E (tx(J(Z))) = «tx (0)) = t;(0) = 1.
==> ) If socp(Rad R) 7É 0 then 0 -> socp (R) -~R ---~R/ socp(R) -~0 does not split .For if it split, then taking a simple submodule S of Rad R we have that the monomorphisms S C ) socp(Rad R), socp(Rad R) C ) socp (R) and socp(R) C-3 R are splitting ; so its composition also splits .So we would have that R = SED K, where RK is a maximal ideal of R, but this is impossible (S <_ Rad R <_ K ==> S n K = S Y 0) .Hence Goldman's torsion theory does not split, and a fortiori, does not centrally split .
Corollary 4. If R is a commutatíve perfect ring, then Goldman's torsion theory centrally splits.
We should note that the preceeding proof does not apply for non commutative right perfect rings, because socp(Rad R) is not necessarily a right semisimple module .
From Theorem 3 .1 of Raggi & Ríos [11], we have that for a right perfect ring, Goldie's torsion theory rG is a TTF torsion theory generated by the left singular simple modules and cogenerated by the left projective simple modules (in fact the preceeding statements hold when R is left semiartinian ring).
In the following theorem we will denote SI the class of the left injective simple modules and by Sp the class of left projective simple modules.Theorem 17.If R is a right perfect ring satisfying socp(RadR) = (0), then are equivalent: i) X, = TG, where X, denotes the least element of [T] E R-tors/^'F.ii) SI = Sp.
Proof.i)==> ii) Sp C SI follows from the part ) of the proof of Theorem 16 .Let RS be a left injective simple module .We want to prove that it is projective .Let us observe that since R is right perfect, then R/ RadR is semisimple, so that RM is semisimple iíf Rad R M = 0 .Therefore every direct product of simple modules is semisimple .As a consequence, using Theorem 18, we get that X(S) belongs to [X]F.For if M E FX(s), then 3m >-+ Sx for some set X, and as SX is a semisimple module.But on the other hand, M is injective, as it is isomorphic to a direct summand of the injective module SX .
Thus, X(S) E [X]F, and therefore X(S) > X, = rG.Then we have that S is Goldie torsion free, which is cogenerated by the left projective simple modules.Hence 3 0 :~f : S -+ U, where U is a left projective simple module .Since f must he an isomorphism, we have that S is a projective module .Therefore SI C Sp, and hence SI = Sp .
ii)==> i) Since TG is cogenerated by the left projective simple modules, we have that every TG-torsion free module is semisimple, since it is (isomorphic to) a submodule of a direct product of simple modules (this product is annihilated by Rad R).But a -rG-torsion free module is an injective module, since it is a direct summand of a product of projective simple modules, and such a product is injective by the hypothesis that all projective simple modules are injective modules .Since every TG-torsion free module is injective, TG E [X)F by Theorem 3.
Analogously, if T C-[X]F let us take E an injective module which cogenerates T; Le., T = X(E) .By another use of Theorem 3, we get that E is semisimple .Now, if RS is a simple submodule of E, it has to be injective.Because S is an injective module, S is also projective by hypothesis .Therefore it is TG-torsion free.So, E E FG, since E is a direct sum of TG-torsion free modules .But E E F G ==:> T = X(E) > TG; so we have that TG = X= .
Corollary 5.If R is a quasifrobenius ring (QF-ring , then X= = rG. Proof: R is right perfect and the class of projective modules coincides with the class of injective modules .Moreover, socp(RadR) = 0: if RS _< RadR was a projective simple module, then as S had to be injective, S would be a direct summand of R. Consequently, S = Re _< Rad R, with e = e2 , this is impossible .We conclude using Theorem 17.
is in KIF .So R-tors/^'F has only the two elements [X]F = {X}, and [fF = R-tors\{X} .Moreover [~] has a maximal member: X(R) = TL, Lambek's torsion theory.v) For a stable torsion theory T the following statements are equivalent: a) Rtr(R) x S, where S is semisimple artinian .b) T E [X]F- to: c) `dN E FT , N is an injective semisimple module .Proof.a) b) (See [111), b) t=> c) follows from Theorem 3. vi) For a left semiartinian ring are equivalent a) -rG E [X] ( TG denotes Goldie's torsion theory) .b) R = TG(R) x S, where S is semisimple artinian.c) rG centrally splits .d) T o is stable .Here -ro denotes Goldman's torsion theory ; Le., the torsion theory generated by the projective semisimple modules.Proof-b) c) d) (See [111) .a)~! b) follows from Remark v).vi¡) If R is right perfect ring, then the above conditions are also equivalent e) soc p (Rad R) = 0 (See Theorem 18) .Here socp denotes the projective socle, and Rad R denotes the Jacobson radical .

Theorem 4 .
Let T be an element of R-tors .Then [7-I F is elosed under finite meets.Proof. .Let us suppose that -rl -F r2 -F r .By the observation after Theorem 2 we have that Arl C ,,4,,,\, (ri A T 2 < r2 ) .Now, let us consider the diagram 0

Theorem 5 .
If R is a left perfect ring, then[r]  is closed under taking arbi- trary meets, * E R-tors .Proof: Let P' E PT and let K' -; P(N) 8 P(N) :: P,(N) P' 0 : L-1-, M -' --) N ) 0 be a diagram with L E FA[,]  .Let 0 -) K(N) -) P(N) -) N -) 0 and 0 -) K, (N) -) P, (N) -) N -) 0 be a projective and r-codivisible covers, respectively.Then 3 a : P' -) PT (N) such that in the first square, we get that 30 : P,(N) ---) M such So we have that in the diagram commutes (because P' is r-codivisible and 0 -) K, (N) ) P, (N)3 N --) 0 E A,.), where 7r' is the epimorphism provided by the projectivity of P(N), and u is the morpHsm obtained from the universal property of kernels .Moreover, by Theorem 1, we have that K' E T,,, Va E [-r].Hence we get K' E TA,~,, .As L E FA,,,,, we get u = 0.But then, given the commutativity that Q o s = 7r' .the square and the top triangle commute ; Le., ir o s = p o 7r' = p o /3 o s.But as s is epi, we have that 7r = p o /i; Le. the bottom triangle is also commutative .Summarizing, we have the following commutative diagram from which we get that P E PA,,, .Hence P r C P"IT, and then AA,,, C .,4r.But A[T] < -r ==> A"IT1 C ,,4T (observation after Theorem 2).Hence A,,~, = Ar and so Aj,]u ^'F -r.So we have proved A [T] E [T] and this is suf~cient for seeing that [T] is closed taking under arbitrary meets ({Ta} C_ [T] ==> n[T] < A{Tv.} < TQ and henceAr« C AA(ra} C AA[r] = Al j-9 -_ Theorem 6 .If R is a lef perfect ring, then [T] is closed under arbitrary joins .Proof.It's enough to prove that V[T] E [T] .Let where the row is a T-codivisible cover of M and where P' is a V[T]-codivisible module .By Theorem 2 we have that L E F,,Vo, E [T] ; hence L E n[,1F o = Fv[rj .So, (*) belongs to Av[rj, and consequently 3«: P' -> P r such that po ix = a.Hence P' E Pr and so Pv[rl C Pr, which is equivalent to saying that Ar C A,[r] .On the other hand, T < V[T] Ar D A,[rj .Then Ar = Av[r] and so V[T] E [T] .P' 0 .L T -----------Pr-p M -. 0 From the two preceeding theorems we get at once: Theorem 8.If R is a lef perfect ring, then : i) T* = x {K,(M)IO --> K,(M) -> an A,-codivisible cover, M E R-mod } .ü) T* = j{K(P.,(M))IO--) K(pr(M)) is a projective cover of P,(M), whereP,(M) M E R-mod} .Proof.First, let us observe that the sequence LATTICE R-TORS 23 Theorem 7. R Lef perfect ==> [T] is a complete sublattice of R-iors, VT E R-tors .

where
0 -> K(P'(M)) -> P(M) --> P'(M) -i 0 P, (M) -> M -> 0 is -, P(M) ---, P'(M) -> 0 is a T-codivisible cover of M, in ii) comes from the diagram and the column are exact, the rows are the projective and the ,r-codivisible covers of M, respectively, and the R-morphism P(M) --> P, (M) is given by the projectivity of P(M).i) By the note after Theorem 2, we have that K,(M) E F o Vo, E [r] ; so X{K,(M)IM E R-mod} >_ r* .Hence X{KT (M)IM E R-mod} >_ r* .It would be enough to see that X{KT(M)IM E R-mod} E [T] and for this it would be enough to see that PX{K,(M)lMER-mod} C PT*-K E F,-, then by taking a T-codivisible cover of M we get the (M) E FX{K,(M)lMER-mod}, 3ú : P -~P,(M) such that 7r o ix = a.Inasmuch as K E F T * C F T , 3 ax : PT (M) -> L such that p o a = 7r, hence So, let P E P T» and is commutative, too.Now, p o (ix o ix) = a and then P E P,.» .So Px{K,(M)1MER-mod} C-P,. .Hence r* <_ X{KT(M)IM E R-mod} and hence r* = X{KT (M)IM E R-mod} .ii) By Lemma 1, we have that K(PT(M» E TA hence j{K(P,(M)IM E R-mod} < r* = A[-r].To get the converse inclusion, it is enough to see that Pr» C P¿(K(P,(M)1MER-mod} " such that K E FI{K(P,(M)1MER-mod} .Let us take 0 ) K(PT(M)) ---) P(M) -> P, (M) ) 0 as in the statement .Then KT (PT(M)) E T^[Tp In the diagram 0 Tr is given by projectivity of P(M), and ,Q is the restriction of fr to K,(PT(M)), we have that /l = 0, inasmuch K E Ff{K(P,(M)1MER-mod} .Then, by the universal property of cokernels, we have that 3 P : PT(M): --+ L such that commutes .But as P(M) --~P,(M) is epic, we have that PI(M) with P E P, .and KT (M) E Fo (Va E [r]) imply that K,(M) E F, ., and so 3-y : P -; PT (M) such that r o y = a.But then commutes .PHence P E PI{K(P* (M))IMER-mod} Thus, Pr.C P¿(K(P,(M))IMER-mod} .SO we get T* = 1{K(PT (M))1 M E R-mod}.

Proof. Straightforward . Theorem 11 .
If R is a left perfect ring, all of whose torsion free classes FT. are also torsion classes (i.e. each FT is closed under taking factors), then R enjoys ¡he properties of Lemma 2.Proof-We will prove that 1* V T = T*, * E R-tors.As t* _< T'*, we have that 1* V T <_ T * (by Theorem 9 we have that 1* = X(RadR) ; T * _ X(Rad R/t,(Rad R)).The hypothesis that Fr is closed under factors RadR/t r(RadR) E FI. ; hence T * > 1* ) .It remains to prove that 1* V T cannot be different from T* .If it was, then 30 qÉ M E TT. n Fe. v, = TT.n FI. n Fr.And as T * = X(Rad R/t r(Rad R)) (Theorem 10) we have that HOMR(M, E(Rad R/t,(Rad R)) = 0 (*)But as M E F¿. and 1* = X(Rad R) (Theorem 9) we have that 3ú : M»-> (E(Rad R)) X , monomorphism for some set X. Hence 3 x E X such that pxu(M) q£ 0, where p,, : (E(Rad R))X -> E(Rad R) is the canonical projection.Hence, in view of (*), we have that u(M) C (t,(E(Rad R»)' .For if this were not true, 3y E X such that py (u(M)) 91 -t,(E(Rad R)) and hence M NU ) E(Rad R)/tr (E(Rad R)) is not the zero morphism .But E(RadR)/t r (E(RadR) E FT. and M E TT. and so HOMR(M, E(Rad R)/t,(E(Rad R)) = 0. -This is a contradiction .