GENERATION OF PROPER CLASSES OF SHORT EXACT SEQUENCES ALl PANCAR

The generation of proper classes of short exact sequences of modules by subclasses is 
considered. The class generated by two proper classes is studied by means of some operations between 
these classes. These operations are investigated in details for classes of short quasi-splitting, torsion-splitting 
and pure exact sequences of abelian groups.

By module we mean a unitary left Romodule, R being an associative ring with tdentty and by group we mean a Z-module, i.e. an abelian group.
Let be a class of short exact sequences of modules.If a short exact sequence E:0 A B >C 0 (1 1) belongs to/, ot is said to be an R-monomorphism and 13 an R-epimorphism.A short exact sequence E is determined by each of the monomorphism t and epimorphism 15 uniquely up to isomorphism.In the case of necessity, without loss of generality, we shall regard any monomorphism to be an inclusion map B stands for the end of the proof.We use the terminologies and notations of Fuchs [1] and Mac Lane [2].DEFINITION.The class is said to be proper, if it satisfies the following conditions (see Buchsbaum  [3], Mac Lane [2], Sldyarenko [4]): (P-1) Along with any short exact sequence contains every one isomorphic to it.(Po2) contains all splitting short exact sequences.
(P-3) The composite of two -monomorphsms is an -monomorphism if ths compomte ss defined.
(P-4) If c, are monomorphisms and Io is an -monomorphism, then c s an -monomorphism.
Short exact sequences of the form (1.1) from give the subgroup E xt(C,A) of the group of extensions E xt(C,A).Throuout this paper we shall write Ext(C,A) instead of E xt(C,A) DEFINITION.Let be a class of short exact sequences.The least proper class containing is said to be generated by and denoted by <>.
Since the intersection of any family of proper classes is proper, we have evidently <> _ is a proper class}. (1.2) The noUon of a generated proper class is a generalization of projectvely (mjectively), coprojectively (coinjectively) generated, and other, proper classes (see StenstrOm [5], Alizade [6]).In A. PANCAR this paper some properties of the class <8> are gven in terms of 8 and the structure of this class ts investigated by means of some operations between subclasses. 2. GENERATED PROPER CLASSES.DEFINITION.A module A is called -coprojective (-coinjective) if every short exact sequence ending (beginning) at A belongs to .
On the other hand, since E t, for any module A we can write the following exact sequences: >Hom(F,A) ;Hom(K,A) Ext(P,A) Ext(Fo,A) Hom(K,A} E xt(P,A) E xt,(Fe,A) >... E xtfFr,,A E xt (Fp,A) 0 Fp being a projective module.By 5-Lemma E xt(P,A) E xt (P,A) Then P s an t-coprojective module.Since k() is the least proper class for which every P is coprojective, we have k (tP) _. 1 The dual statement can be proved for coinjectively generated classes.DEFINITION.Let t be a class of short exact sequences.A module A said to be t-projective (-injective), if for every t-epimorphism (t-monomorphism) o" B---C ." Hom(A,B) >Hom(A,C) ('" Hom(C,A) ;Hom(B,A)) (2.3) is an eptmorphism.The class of all t-projeetive (respectively, t-injective) modules is denoted by (t) (resp.t()) (see [4], [5]).
A proper class is said to be projective, if for every module A there is an -epimorphism from an t-projective module P onto A. An injective class is defined dually.(2.4) from t. P being an -projective module.Then <.PROOF.The inclusion <=_ is obvious.On the other hand, let (1.1) be an arbitrary short exact sequence from .Since is a projective class, there exists an -epimorphism '/ P-+B with an -projective P. By (P-3') 13o'/is an -epimorphism At the same time 13o,/is an <>-epmorphism since P is an -projective module.Therefore, 13 is an <>-epimorphism by (P-4').Thus E <. l The proposition dual to propostmn 2.2 can be proved for an injective class t PROPOSITION 2.3.g(8) (<8>).
The proof s dual to that of Proposmon 2.3 3. SOME OPERATIONS BETWEEN PROPER CLASSES.DEFINITION.The proper class tt+l =<tt/> is called the sum of the proper classes and 1 Let us denote the class of short exact sequences whose monomorphisms are ctol3, ot being an -monomorphism and 13 being an l-monomorphism, by toL Define the class .of short exact sequences by the formula We have the followang obvious facts: 1) t+.1+t is a proper class; 2) ,1 3) ol _ 1+ and In Theorem 3.1 we present the situation where ,1 _ ol.
In general tol and t,l need not be a proper class (see Corollary 4.1) But if to.(or ,1) is proper then, clearly, tt+l o. (tt+l a,l) and we can study t-I by means of and 1 (as in [6]).
The following proposition can be derived from Theorem 2.1 of Kepka [8], but we give another proof of this fact which relies on Thoeorem of Alizade [7].
PROPOSITION 3.1.If the class of modules tP is closed under submodules and extensions (see [7], [8]), then k (tP) ' o, 0, ' being the class of all short exact sequences ending at modules from t and '0 being the class of all splitting short exact sequences.
By Prop.3.1 or' can be written as x'=o0, where 0 s a splitting monomorphism and P=D/Imot Therefore 0ol3'=[3 s an -monomorphism.Thus 6=oto[3 and we have the following commutatsve d,agram wth exact rows and columns: The short exact sequence 0 >M ' B' C >0 belongs to I as an image of the short exact sequence 0.
To prove that Ext.(C,A) is a subgroup of Ext(C,A) for every C, A, let E, E_Ext.(C,A).
One can easily show that E1 implies -Eol.Thus E xt. is a subfunetor of Ext R-rood x R-rood ---> Ab.1 REMARK.Now to prove that ol is a proper class it remains only to show that the composite of two o,-monomo'rphisms is an o2-monomorphism (see Nunke  [9]).By means of ths fact we have obtained the description of the smallest proper clasps with given classes of coprojcctive and omjeetive modules as a "composite" of coinjectively and coprojectively generated classes, which will soon be pubhshed Since _oA and d_co2, Lemma 3.2 gives the following inclusion.THEOREM 3.1.If the class P of modules is closed under extensions and submodules (see [7], [81), then ,2_o2 for k (P) and for every proper class 4. SOME EXAMPLES l THE CATEGORY OF ABELIAN GROUPS.Let 0 be the class of all splitting short exact sequences, be the class of all pure short exact sequences and 9 be the class of all torsion splitting short exact sequences (see Fuchs ]).For a proper class put {E for some nZ (n0) nE}, (4 1) (see Ahzade [6]).
Class 0 of all short quasi-splitting exact sequences was introduced and studied by Walker C.P. [10] Classes and , were studied by Hart in [11], where these classes were denoted by Clearly, +'(0for every proper class .
The endomorphsm n:AA can be represented in the natural way n=czoo, where o.A-,nA s pimorphlsm and ct:nA-A is inclusion map We have the following commutative diagrams with exact rows and columns: A/nA is a bounded and hence 0-coprojective group.Therefore the short exact sequence 0 >nA A >A/nA >0 belongs to 0 and since 0_?, it belongs to g,e., a s a ?-mono-morphism.Since nE_;, monomor-phism.Hence, by (P-4), v is a ;?-monomorphism.Thus E'?.
A[n] Is a bounded and therefore 0-coinjective group.Hence the short exact sequence 0.

>A[n]---->B
>B, >0 belongs to 'o and since ,0_, it blongs to , e. y s a -epimorphsm.Since E', 5 is a ;-epimorphism.By (P-3') 0=8oy is a -epimorphism.Therefore E? and we have _:.I DEFINITION.For a subgroup A of B let ={ bB: mbA for some mZ,(m0)} LEMMA 4.1.Let B be the maximal divisible subgroup and T(B) be the torsion part of a group B. PROOF.The inclusion Bd+TCB)_B is obvious.
Let B=BdK, K<B and let b=a+k be any element from B (aeBd, keK).Then nbeBd for some he0.Since Bd is divisible, nb--na, for some a, eBb.Therefore na+nk=nb--na,.By directness of the sum BK, nk=0 and hence keT(B).So b=a+keB,+T(B) Now we take B=Ext(C,A) for any groups C,A, then Bd=EXts(C,A) by Prop. 58. for each group C,A, i.e., 9=, This equahty together with Theorem 3.1 (note that3 k(#), being the class of all torsion free groups (see [7]) gives us the following theorem.THEOREM 4.2.*A0 oA +o:,.So every ,-monomorphism is a composmon of an 0-monomorpMsm and a monomorphism. 1 For the class $ the situation s worse.LEMMA 4.2.There exists a reduced cotorsion group C, Uim's subgroup of which s torsmn flee and algebraically compact group PROOF Let J be the group of p-adm numbers.The p-basic subgroup B=<b> of J, is somorphm to the group Z of integers and Jp/B is a divisible group (see ch.20 and 32 in Fuchs [l]).Let us construct an extension K of B such that the first UIm's subgroup K of K is equal to B. Define K by K/B is isomorphic to a direct sum of cyclic groups, hence (K/B)' =0 and we have K' c_B.On the other hand, the inclusion B_ K is obvious and thus K=B.
Let us complete the diagram (4 9)   to a pushout diagram.Then we have the following commutative diagram with exact rows and columns: Since ct s a monomorphism, or' is also a monomorphism and Jp/BD/K, K/B=D/Jp (see Th 10.2 [1]).Now let us prove that D=Jp.Since (D/Jp)--(K/B)=0 we have D c_Jp.On the other hand let xJ and nZ (n,0) be arbitrary.Since Jp/B is divisible, there is x, Jp such that x-nxt B B=K t, 472 A. PANCAR therefore there is an element kK such that x-nx,=nk.Then x=n(x,+k)nD Thus we have xD' and D =J Since D/Jp is isomorphic to a direct sum of finite cyclic groups, it can be embedded into a direct product of these cyclic groups, i.e., into a reduced algebraically compact group M D/Jp M. By Theorem 51.3   13": E xt(M,Jp) E xt(D/Jp,J) Jp=DIcC '_ Since M is a reduced algebraically compact group, we have (C/Jp)=-M=0 by Prop.54.2 [1].Then C _Jp. Hence we have C'=Jp and C is a reduced group.By Prop. 39 4 [1]Jo s an algebraically compact group.Since J and M are algebraically compact and hence cotorsion groups, by ch 54(D) [1] C is a reduced eotorsion group.PROOF.Take an element a n A of the reduced group A. Since A= n C, there s bC such 0nZ that nb=a.Obviously b ,.If bA+T(C), one can write b=a,+t with a, cA and tT(C).Then a=nb=nat+nt and hence nt=a-na AcT(C).Since A is torsion free, nt=0 and we have a=na nA But ths contradicts the assumption.Thus b , \(A+T(C)).[ The following theorem demonstrates that for proper classes t and I the class t.l need not be proper.THEOREM 4.3. ,;e .PROOF.According to Lemma 4.2 there is a reduced cotorsion group C, Ulm's subgroup A=C of which s reduced, torsion free and algebraically compact.Ext(QF/C)+Ext (Q/Z,C) Ext^(Q/Z,C) Thus ,0;e;. 1 COROLLARY 4.1.The class *0 is not proper PROOF.Since E xt(C,A) and E xt^(C,A) are subgroups of E xt^(C,A) for any group C and A, ewdently 4, 0 -;-Since ;=+/ is the least proper class including and 0, if ,, Were a proper class we should have -*0 and hence ,=,.But this contradicts Theorem 4.3.
ACKNOWLEDGEMENT.The author wishes to express his gratitude to his referee for helpful criticism.

( 4 .
11) s an epmorphism.Therefore there is an extension C of D such that the following diagram with exact rows and columns s commututive:

LEMMA 4 . 3 .
If the Ulm's subgroup A=C' of a reduced group C is torsion free and different from zero, then A+T(C) ;e , ,(4 13)T(C) being the torsion part of C.