8. QUOTIENT DIVISIBLE MODULES. E. L. Lady. April 17, 1998

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1 1 8. QUOTIENT DIVISIBLE MODULES E. L. Lady April 17, 1998 Now that we have seen how a p-divisible module can be understood by looking at d(ŵ G), it is tempting to wonder if this insight can be used globally. The following conjecture seems plausible: If ϕ: QG QH and ϕ(d(ŵp G)) d(ŵp H) for all primes p, thenϕis a quasi-homomorphism from G to H. From the Kurosh Theorem (Theorem 6.*) we see that this conjecture can be restated as a local-global principle: If ϕ: QG QH is such that ϕ is a quasi-homomorphism from G p to H p for all prime ideals p, thenϕmust be a quasi-homomorphism from G to H. In fact, over any dedekind domain with an infinite number of prime ideals this conjecture is false. example 8.1. Let A be the rank-one module from Example 2.*, i.e. the submodule of Q generated by p 1 for all prime ideals ideals p. Let ϕ: QA Q be the identity map on Q. Thenϕis a quasi-homomorphism from A p to W p for all primes p since A p W p and QEnd W p = Q. Butϕis not a quasi-homomorphism from A to W since t(a) t(w ) and thus Hom(A, W )=0. In [BP] Beaumont & Pierce defined a class of modules G which are precisely those for which the above conjecture is true. These modules are called quotient divisible modules. For the moment we will make a provisional definition: definition 8.2. We will say that a finite rank torsion free module G has the localglobal quasi-homomorphism property if for all torsion free modules H and all ϕ: QG QH, ϕ QHom(G, H) ϕ QHom(G p,h p ) for all primes p. (If we, as usual, identify QHom(G, H) with a submodule of Hom(QG, QH), then we have that G is quotient divisible if and only if QHom(G, H) = p QHom(G p,h p ).) To motivate the definition of a quotient divisible module, think first about why one might expect all modules to have the local-global quasi-homomorphism property. The plausibility of the conjecture above arises from the equations G = p G p and H = p H p. When we consider Example 8.2 in that light, we realize that the reason it

2 fails is that although A p W p for all p, the intersection A p is as it were skewed. So what we need to look for is a condition that guarantees that all the localizations G p somehow line up. To try to see what it would mean for the localizations G p to line up, consider the following: let F be an essential projective submodule of G. Then (G/F ) p G p /F p is q-divisible for each prime q different from p. Furthermore, it follows readily from Proposition 1.* that (G p /F p )[p] has finite dimension over W/p. We can thus easily conclude that (G/F ) p is the direct sum of a divisible module and a finite length p-primary module M,whereM (G/F )[p ]. If we now let F F be the pre-image of M in G (i.e. M = F /F ), then F is finitely generated and thus projective, and G/F is divisible. One is thus tempted to say: Without loss of generality, we may assume that (G/F ) p is divisible. If true for every p, this would mean that G/F would be divisible. However this is fallacious, because in the example above, for instance, where A is the submodule of Q consisting of fractions with square-free denominators, there does not exist a finitely generated submodule F A with A/F divisible. The fallacy arises because the submodule F chosen above depends on the prime p, and in general no single F will work for all primes p. The existence of an essential finitely generated submodule F of G such that G/F is divisible is precisely the condition we are seeking that explains what it means for all the localizations G p to line up, so that the intersection G = p G p is not skewed. definition 8.3. A finite rank torsion free module G is called quotient divisible if there exists a finitely generated essential submodule F G such that G/F is divisible. PROPERTIES OF QUOTIENT DIVISIBLE MODULES. Recall that according to Proposition 2.*, if F and F area finitely generated essential submodules of a finite rank torsion free module G then the quotients G/F and G/F are torsion modules differing only by a finite length summand. One can thus associate with G an equivalence class of torsion modules G/F which is called the Richman type of G. Obviously G is quotient divisible if and only if its Richman type contains a divisible module. proposition 8.4. (1) Direct sums, pure submodules and torsion free homomorphic images of quotient divisible modules are quotient divisible. (2) If G is quotient divisible and H is quasi-isomorphic to G, thenh is quotient divisible. (3) Projective modules are quotient divisible. (4) If W is semi-local then every finite rank torsion free W -module is quotient divisible. (5) If R is a quasi-separable semi-prime ring then R is a quotient divisible W -module. (6) If R is a splitting ring (not necessarily with finite rank) then every R-split module is quotient divisible. 2

3 proof: (1) It is clear that the direct sum of two quotient divisible modules is quotient divisible. Now let H G and let F be a finitely generated essential submodule of G such that G/F is divisible. Then H F is a finitely generated submodule of H and H/H F is a pure submodule (???) of the divisible module G/F hence is divisible. Thus H is quotient divisible. Furthermore (F + H)/H is an essential submodule of G/H and (G/H)/((F + H)/H) G/(F + H), which is a homomorphic image of the divisible module G/F, hence is divisible. Thus G/H is quotient divisible. (2) We may suppose G H. Then by Proposition 3.* H/G has finite length. Let F be a finitely generated submodule of G such that G/F is torsion divisible. Then the divisible module G/F must be a summand of H/F : H/F G/F H/G. Ifwenowlet F Fbe such that F /F is the component of the above direct sum isomorphic to H/G, then F is finitely generated (both F and F /F are) and H/F G/F, hence H/F is divisible. Thus H is quotient divisible. (3) Clear. Choose F = H. (4) For each prime p, choose a finitely generated submodule F p of G such that G p /F p is divisible. (Note quite right!) Let F be the submodule of G generated by all the F p. (5) By Proposition 7.* and Proposition 7.* it suffices to consider the case when R is an integral domain D. SinceQD is separable over W, the integral closure of D in its quotient field QD is a finitely generated D-module, hence is quasi-equal to D. Thus there is no loss of generality in supposing D integrally closed. Now let W be the integral closure of W in QD. By the separability of QD, W is a finitely generated essential W -submodule of D. Furthermore W is a dedekind domain and D is a rank-one W -module as well as a subring of QD. By Proposition 2.*, then, D/W is a divisible W -module, hence a fortiori divisible as a W -module. Therefore D is quotient divisible. 3 corollary 8.5. If G and H are quotient divisible submodules of a Q-vector space V, then G H and G + H are also quotient divisible. proof: G H and G + H are isomorphic to a pure submodule and a homomorphic image of G H. Among modules of arbitrary finite rank, the quotient divisible ones are special in the same way that the rank-one modules with idempotent type are special among the class of all rank-one modules: proposition 8.6. (1) A rank-one module A is quotient divisible if and only if t(a) is idempotent. (2) If G is quotient divisible then T(G) and IT(G) contain only idempotent types. (3) A Butler module G is quotient divisible if and only if T(G) contains only idempotent types.

4 proof: (1) ( ): If t(a) is idempotent then by Proposition 2.*, A is quasi-equal to a subring of Q. By Proposition 8.4, a subring of Q is quotient divisible. ( ): We may suppose W A. IfAis quotient divisible then A is quasi-equal to A, where A /W is divisible. Hence wlog assume A/W divisible. **** (2) If G is quotient divisible then by Proposition 8.4, every pure rank-one submodule of G is quotient divisible and hence by (1) is idempotent. Thus T(G) contains only idempotent types. (3) ( ): By (2). ( ): If G is a Butler module then by Proposition 5.* G is isomorphic to a homomorphic image of a module of the form A i, where the A i are rank-one modules with t(a i ) T(G). If T(G) contains only idempotent types, then by (1) each A i is quotient divisible. Hence by Proposition 8.4, G is quotient divisible. 4 example 8.7. There exist modules G such that T(G) andct(g) contain only idempotent types but G is not quotient divisible. proposition 8.8. If G is a quotient divisible module and H is a submodule of G with rank H =rankg and p-rank H p-rank G for all primes p then H is quasi-equal to G. proof: Since H is essential in G the hypothesis implies that p-rank H = p-rankgfor all p. **** proposition 8.9. A quotient divisible module G is projective if and only if p-rank G =rankg for all primes p. proof: ( ): By Proposition 1.*. ( ): Let F be an essential free submodule of G. Then p-rankf =rankf = rank G =p-rankg for all primes p. Therefore by Proposition 8.8, G is quasi-equal to F. Therefore G is projective by Proposition 3.*. proposition (1) If G and H are quotient divisible then so is G H. (2) If H is quotient divisible then so is Hom(G, H). proof: (1) Let F 1 and F 2 be essential free submodules of G and H respectively such that G/F 1 and H/F 2 are divisible. Then F 1 F 2 is an essential submodule of G H and (G H)/(F 1 F 2 ) is a pure submodule (????) of G/F 1 H F 2, which is divisible. (2) Let F be an essential free submodule of G. Then G/F is torsion, so Hom(G/F, H) = 0. Therefore there is an exact sequence???? Hom(G/F, H) Hom(G, H) Hom(F, H). We can now fulfill a promise from Chapter 7:

5 5 proposition Let F be a finite separable extension of Q and G a finite rank torsion free module. Then G is isomorphic to an essential subring of F if and only if it satisfies the following three conditions: (1) rank G =[F:Q]. (2) QEnd G contains a field isomorphic to F. (3) G is quotient divisible. proof: ( ): It was shown in Theorem 7.* that if G is an integral domain with quotient field F then conditions (1) and (2) are satisfied. Furthermore G is quotient divisible by Proposition 8.4. ( ): **** THE LOCAL GLOBAL PRINCIPLE. proposition Any G with the local-global quasi-homomorphism property (Definition 8.*) must be quotient divisible. proof: In fact, for any G let F be a finitely generated essential submodule of G and let G F be such that G /F = d(g/f ). Then clearly G is quotient divisible. Furthermore (G/G ) p has finite length for each prime p (as follows from Proposition *), and thus for each p by Proposition 3.* the identity map 1 QG is a quasi-homomorphism from G p to G p inverse to the inclusion map G p G p Thus if G satisfies the local-global quasihomomorphism property then G is quasi-isomorphic to G. Hence by Proposition 8.* G is quotient divisible. We will now show that G has the local-global quasi-homomorphism property if and only if G is quotient divisible. First observe the following: proposition Let F be an essential finitely generated submodule of G. Suppose that G has the following property with respect to F : Whenever H is a torsion free W -module and ϕ QHom(G p,h p ) for some prime p and ϕ(f ) H,thenϕ Hom(G p,h p ). Then G has the local-global quasi-homomorphism property. proof: Let ϕ: QG QH and suppose ϕ QHom(G p,h p ) for all primes p. Since F is finitely generated, there exists 0 w W such that wϕ(f) H. The hypothesis then yields wϕ Hom(G p,h p ) for all p, sothatwϕ(g) H p = H. Thus wϕ Hom(G, H), so that ϕ QHom(G, H). This shows that G has the local-global homomorphism property. Now the clincher:

6 proposition If F is a submodule of G such that G/F is divisible, and ϕ QHom(G, H) for some torsion free H,thenϕ Hom(G, H) ϕ(f ) H. proof: ( ): Clear. ( ): If ϕ(f ) H then ϕ induces a homomorphism ϕ: G/F QH/H. Ifalso ϕ QHom(G, H) thenwϕ(g) H for some w 0 W,sothatw ϕ(g/f )=0. But ϕ(g/f ) is divisible since G/F is divisible. Since a non-trivial divisible module is faithful, this means ϕ(g/f ) = 0, i.e. ϕ(g) H. Thus ϕ Hom(G, H). To summarize: theorem G has the local-global quasi-homomorphism property if and only if G is quotient divisible. 6 proposition If G and H are quotient divisible modules with rank G =rankh and p-rank G =p-rankh for all primes p and ϕ: G H is a monomorphism, then ϕ is in fact a quasi-isomorphism. proof: Your author finds Proposition 8.14 intruiguing, especially when G has a free submodule F with G/F divisible, as is the case, for instance, when G is quotient divisible and W is a principal ideal domain. In this case, let f 1,...,f r be a set of generators for F.Then these are constitute a finite set of test elements, as it were, for a quasi-homomorphism. Namely, according to Proposition 8.14 if ϕ QHom(G, H) thenϕ(g) Hif and only if ϕ(f i ) H for all i. From this we get an interesting corollary. In general, given finite rank torsion free modules G and H it is fairly difficult to get much information about Hom(G, H). When G is quotient divisible, we have the following result: proposition If G is quotient divisible and rank G = r, then for any torsion free H, Hom(G, H) is isomorphic to a pure submodule of H r 1 ah, for some ideal a of W. If there exists a free submodule F of G with G/F divisible (in particular, if W is a principal ideal domain), then Hom(G, H) is isomorphic to a pure submodule of H r. proof: Let F be a projective submodule of G such that G/F is divisible. According to Proposition 8.14 Hom(G, H) =QHom(G, H) Hom(F, H), where we have identified QHom(G, H) as a subspace of QHom(F, H) using the exact sequence Hom(G/F, H) =0 Hom(G, H) Hom(F, H). Thus Hom(G, H) is isomorphic to a pure submodule of Hom(F, H). By Proposition 0.* F = F 1 I where F 1 is free and I is an ideal of W. Then by Proposition 1.* Hom(F, H) Hom(F 1,H) Hom(I,H) H r 1 I 1 H. But I 1 a for some ideal a W.IfFis free, then Hom(F, H) H r.

7 7 proposition Suppose that there exists a free submodule F G such that G/F is divisible. Let Hom(G, H) be embedded as a pure submodule of H r as in Proposition Let ϕ Hom(G, H) correspond to h 1,...,h r H r under this embedding. Then ϕ is monic if and only if h i 0for all i. Furthermore if H is the submodule of H generated by the set {h 1,...,h r }, then the reduced part of H/H is isomorphic to H/ϕ(G). proof: **** remark It would seem that in most cases, a quotient divisible module G would have a free submodule F such that G/F is divisible. For if F is a finitely generated essential submodule of G with G/F divisible then F is a direct sum of ideals. Consider a rank-one summand I of F. If I is not a pure submodule of G then since I is isomorphic to an ideal of W it would seem plausible that there would exist a cyclic module I with I I I. If one can do this for each rank-one summand for each rank-one summand I of F,thenletF be the direct sum of these cyclic modules I. Then F is free and F F,sothatG/F is a homomorphic images of G/F and hence is divisible. Clearly there are cases where this will not work, however. For instance if G is projective but not free and F is an essential free submodule of G then G/F is non-trivial with finite length, hence not divisible. corollary If G is quotient divisible and A is a rank-one module and t = t(a), then Hom(G, A) is a t-projective module. proof: By Proposition 8.17 Hom(G, A) is quasi-isomorphic to a pure submodule of the t-projective module A r. By Proposition 4.* quasi-pure submodules of t-projective modules are t-projective. When t is idempotent, by Proposition 4.* Corollary 8.20 is true without any hypothesis on G. However for an arbitrary type t, and especially when t is locally trivial, this result is in surprising contrast to what we will later see in Proposition 9.*. EXISTENCE THEOREM. theorem Let U be a finite dimensional Q-space. For every prime p of W,let D p be a QŴp-subspace of Ŵ p U. Then there exists a unique (up to quasi-equality) quotient divisible W -module G such that (1) QG = U. (2) For every prime p, d(ŵp G) =D p.

8 proof: Let F be an essential free submodule of U. For each prime ideal p, choose a pure submodule F p of F such that F p D p =0and(Ŵp F p ) D p is an essential Ŵ p -submodule of Ŵ p U. (How do we know this can be done?) Let G p = U ((Ŵp F p ) D p ). (The subscript p here does not denote localization.) Let G p = p G p. Then QG = U and for each prime p, d(ŵp G) =D p since ****. Furthermore, we claim that G/F is divisible, which will show that G is quotient divisible. In fact, it suffices to prove that for each prime p, Ŵ p (G/F ) is divisible (why?). But Ŵ p (G/F ) (Ŵp G)/(Ŵp G) D p, a divisible module. Thus G/F is divisible and since F is an essential free submodule of G, this shows that G is quasi-divisible. Now if G is another quotient divisible module with QG = U and d(ŵp G) =D p for all p, andifϕ=1 U, then by the Local Global Homomorphism Principle ϕ is a quasi-homomorphism both from G to G and from G to G. Thus G is quasi-equal to G. THE QUOTIENT DIVISIBLE CORE. proposition (1) If G is a finite rank torsion free module then there exists an essential quotient divisible submodule G 0 of G. (2) G 0 is unique up to quasi-equality. (3) G/G 0 is a reduced torsion module. proof: (1) Let F be an essential free submodule of G and let G 0 be the inverse image in G of the divisible submodule of G/F. Thus G 0 /F is divisible so G 0 is quotient divisible. (3) G/G 0 (G/F )/(G 0 /F ) and the latter is a reduced torsion module since G 0 /F is the maximal divisible summand of the torsion module G/F. (2) Now suppose G 1 is also an essential quotient divisible submodule of G. Then by Corollary 8.5 G 0 G 1 is quotient divisible. Therefore wlog we may suppose that G 0 G 1 (Explain!). NowletFbe a finitely generated submodule of G 0 such that G 0 /F is torsion divisible. Then by Proposition 2.* G 1 /F is isomorphic to the direct sum of a divisible torsion module and a finite length module T. ThenTis not faithful, so there exists w 0 Wwith wt =0. I.e. wg 1 /F G 0 /F.SinceF G 0 G 1 it follows that wg 1 G 0.ThusG 1 is quasi-equal to G 0. 8 definition The quotient divisible core of G is defined to be an essential quotient divisible submodule of G. Proposition 8.23 thus says that every finite rank torsion free G has a quotient divisible core, and the quotient divisible core is unique up to quasi-equality. In analogy to the construction of the category of finite rank torsion free modules under quasi-homomorphisms, it is tempting to construct a category C in the following way:

9 9 TheobjectsofC are finite rank torsion free modules. For two modules G and H, Hom C (G, H) is defined to be the set of linear transformations ϕ: QG QH such that ( p) ϕ p QHom(G p,h p ). Unfortunately (or fortunately, for those who wish to discourage the proliferation of categories), the category C doesn t really give us anything new: proposition If ϕ Hom(QG, QH) then ϕ is a morphism in C if and only if ϕ restricts to a quasi-homomorphism from G 0 to H 0,whereG 0 and H 0 are the quotient divisible cores of G and H. proof: QUOTIENT DIVISIBLE MURLEY MODULES. proposition An indecomposable Murley module is quotient divisible if and only if its inner type is idempotent. proof: Although we have seen that only quotient divisible modules have the local-global quasi-homomorphism property, for Murley modules there is an analogous, slightly weaker, property: proposition Two Murley modules G and H are quasi-equal if and only if QG = QH, IT(G) =IT(H), andd(ŵp G)=d(Ŵp H) for all primes p. proposition If G is a Murley module then End G is a direct sum of quotient divisible strongly homogeneous modules. In particular, if G is indecomposable then End G is strongly homogeneous. proposition Let G be a Murley module. The following conditions are equivalent: (1) G is strongly homogeneous. (2) The quotient divisible core of G is strongly homogeneous and isomorphic to End G. (3) G is a rank-one torsion free (End G)-module. ARNOLD DUALITY (GLOBAL CASE).

10 theorem There exists a contravariant functor on the category of finite rank torsion free W -modules under quasi-homomorphisms such that for every G the following are true: (1) QAG =Hom(G, Q). (2) If γ : G K then A(γ) =γ =Hom(γ,Q). (3) rank AG =rankg. (4) For every prime p, p-rank AG =rankg p-rank G. (5) AG is divisible if and only if G is locally free. (6) A 2 G is naturally quasi-isomorphic to the quotient divisible core of G. (7) For any torsion free W -module H, d(g H) QHom(AG, H). proof: For each prime p, the localization G p of G is Ŵp-split, and therefore it has an Arnold dual A p G p within the category of p-local modules. Furthermore QA p G p = Hom(G p,q) = Hom(G, Q) andifγ:g Kthen γ induces A p (γ p ): A p K p A p G p and A p (γ p )=γp =γ =Hom(γ,Q). By Theorem 8.21, there exists up to quasi-equality a unique quotient divisible module AG such that for every prime p, (AG) p =A p G p and QAG = QA p G p =Hom(G, Q). Furthermore rank AG = ranka p G p = rankg and for each prime p, p-rank AG = p-ranka p G p = rankg p-rank G. Andifγ:G Kthen γ QHom((AH) p, (AG) p ) for every prime p, so by the Local-Global Homomorphism Principle γ QHom(AK, AG). We define Aγ = γ. Clearly this makes A a functor from the category of finite rank torsion free W -modules under quasihomomorphisms into itself. By construction, assertions (1) and (2) hold and (3) is true since rank AG =rankqag= rank Hom(G, Q) =rankg. (4) For any prime p, p-rankag = p-rank(ag) p = p-ranka p (G p ) = rank G p p-rank G p =rankg p-rank G by Proposition 6.**. (6) Let ρ: QG QA 2 G = Hom(Hom(G, Q), Q)begivenbyρ(g)(ϕ) =ϕ(g). By Proposition 6.**, for every prime p, ρ is a quasi-isomorphism from G p to A 2 p (G p)=(a 2 G) p. Thus by Proposition 8.*, ρ is a quasi-isomorphism from the quotient divisible core of G to the quotient divisible core of A 2 G. But by construction A 2 G is quotient divisible. Therefore A 2 G is quasi-isomorphic to the quotient divisible core of G. (5) G is locally free if and only if G p is a free W p -module for each prime p. By Proposition 6.* this is true if and only if (AG) p = A p G p is divisible for each prime p, which is equivalent to the statement that AG is divisible. (7) Note that d(g H) = p d(g p H p ), where the intersection is taken over all primes p. And by Proposition 6.*, for each prime p there is an isomorphism σ p : d(g p H p ) QHom((AG) p,h p ). Finally, since AG is quotient divisible, by Proposition 8.** QHom(AG, H) = p QHom((AG) p,h) p d(g p H p )= d(g H). 10

11 proposition Let W be a finite rank quasi-separable integral extension of W. (1) If L is a W -module then the Arnold dual of L as a W -module is the same as its Arnold dual as a W -module. (2) If G is a W -module then the Arnold Dual of W G as a W -module is W AG. 11 corollary Let D be an integrally closed quasi-separable integral domain and let D be the unique integrally closed subring of QD such that D D is the integral closure of W in QD and D + D = QD. Then D is quasi-isomorphic to the Arnold dual of D. proof: Since D is integrally closed it is a W -module, where W is the integral closure of W in the field QD. By Proposition 8.30, in computing AD we may replace W by W as the ground ring. Thus there is no loss of generality in supposing that rank D =1. Therefore rank AD = 1 and by Proposition8.6, t(ad) is idempotent. Therefore we may choose AD to be a subring of QW. Furthermore, by Theorem 6.*, for each prime p, p-rank AD = 1 if and only if p-rank D = 0. Thus by Proposition 2.* AD is the unique subring of Q such that D AD = W and D + AD = QW = QD. proposition (1) Let H be an essential submodule of G and let γ : H Gbe the inclusion map. Then Aγ : AG AH is a monomorphism embedding AG as an essential submodule of AH. (2) If G 1 and G 2 are essential submodules of a Q-vector space V then A(G 1 + G 2 )=AG 1 AG 2 and A(G 1 G 2 )=AG 1 +AG 2. proof: (1) If ϕ QAG = Hom(G, Q) thenaγ(ϕ)=ϕγ is simply the restriction of ϕ to H.Since H is essential in G, ϕ 0 Aγ(ϕ) 0. Thus Aγ is monic. Since rank AG =rankah, Aγ thus embeds AG as an essential submodule of AH. (2) It suffices to prove this in the local case. Thus we may suppose that G 1 and G 2 are R-split for some splitting ring R. NowQAG 1 =QAG 2 =Hom(QG 1,Q) and likewise (R AG 1 ) = (R AG 2 ). And by Proposition 3.* QHom(G 1 + G 2,R)=QHom(G 1,R) QHom(G 2,R). Thus by Proposition 8.* A(G 1 + G 2 )=AG 1 AG 2.And A(G 1 G 2 )=A(A 2 G 1 A 2 G 2 )=A(A(AG 1 +AG 2 )) = AG 1 + AG 2. remark. There are two ways in which one might want to improve on Arnold duality. (1) One might look for a different way of defining A so that AG is determined up to isomorphism rather than only up to quasi-equality. (2) One might try to redefine A so that AG is only quotient divisible when G is and A 2 G G for all finite rank torsion free G. Unfortunately, though, neither of these is possible.

12 proof: (1) If A were defined on the category of finite rank torsion free modules under homomorphisms and A 2 G G for quotient divisible modules, then Q =EndQ End AQ =EndW=W, a contradiction. (2) Let B be a rank-one module with t(b) locally trivial. Then p-rank B =rankb=1 forall p,sop-rankab=rankb p-rank B =0 forallp. Thus AB is a divisible rank-one module, so AB = Q and thus A 2 B = AQ = W B. GENERALIZED BUTLER MODULES. theorem Let G be a finite rank torsion free W -module. The following conditions are equivalent: (1) G is R-split for some finite rank quasi-separable splitting ring R. (2) G is isomorphic to a pure submodule of a direct sum of quasi-separable integral domains. (3) G is a homomorphic image of a finite direct sum of quasi-separable integral domains. (4) There exists a separable extension Q of Q such that W G is a quotient divisible Butler W -module, where W is the integral closure of W in Q. proof: (1) (2): If G is R-split then G is isomorphic to a pure submodule of R k for some k.butr is a finite product of quasi-separable domains. (2) (1): If G is isomorphic to a pure submodule of D i, it follows from Theorem 7.* and Lemma 7.* that there exists a quasi-separable splitting ring R such that all the D i are R-split. Then G is R-split. (1) (3): If G is R-split then so is its Arnold dual A(G). Apply Arnold duality to the pure embedding A(G) D i to get a homomorphism ϕ: A(D i ) G such that Image(ϕ) isquasi-equaltog. By Proposition 7.* the A(D i ) are quasi-separable domains. Since G/ Image(ϕ) is finitely generated, we can adjoint a finite number of copies of W to A(D i )andmakeϕsurjective. (3) (4): By Proposition 7.* we can find a finite integral quasi-separable extension W of W such that the W D i are all direct sums of rank-one W -modules of idempotent type. Since W G is a homomorphic image of W D i it is thus a Butler W -module. By Proposition 6.* it is quotient divisible. (4) (2): If W G is a quotient divisible Butler W -module then W G is isomorphic to a pure submodule of a direct sum of rank-one W -modules of idempotent type, i.e. a direct sum of subrings of Q. But G is a pure submodule of W G. 12