Divisibility and purity in modules

A ^ . K . YOUSÜFZAI

o ^ ^ CD

Dr/ISIBILITY AÏÏD PURITY UI MODULES

BIBLIOTHEEK TU Delft P 1938 2469

C 639840 17446

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE V/ETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT., OP GEZAG VAN DE RECTOR

MAGNIFICUS IR. H»R. VAN NAUTA LEMKE, HOOGLERAAR IN DE AFDELHTG DER ELEKTROTECmiIEK, VOOR EEN COMISSIE UIT DE SENAilT TE VERDEDIGEN OP

WOENSDAG 27 OKTOBER 1971 TE 14 UUR

DOOR

AFZAL imMAD KEIJi YOUSÜFZAI

MASTER OF SCIENCE

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR

PROF .DR. F. LOONSTR/..

E R R A T A

Contents: Chapter II, read: S-diyisihle groups and S-pure subgroups.

P. 7 line 17, add :r,séR,aé:A,

P.. 8 line

6,

read : A- > A^ — ^ A^ >

— ^ \

^ ..(k:^2)

P.10 line 1, read : the diagram

E: 0 > A ^^^.B -21^ C > 0

E' : 0 ^ A ' ^ ^ B'-^-> C' > 0

P. 11 line 21, read: modulo T-r,(C,A), module

/_ .\

R^

P. 12 line 6 , rea,d: If E: 0 ^ A ^ B — ^ C ^ 0

P . 13 l i n e 8 , r e a d : R-homomorphism <P :A^ ^ B^ s u c h t h a t t h e d i a g r a m s

A. — i - ^ A

p. 13 Lemma 1.6 line 5, read: 0 ^ A — ^ B - ^ C > 0

"X' 7*^ -S^r

P, 15 Lemma 1.7 last line, read: 0 ^ A •> B ••'-') C

P. 22 line 5» read: A is an S-extension of D ( A ) .

P. 25 line 2 (d), read: o ^

^ - ^ C/^^^-^ > 0

P. 26 Proof of Theorem 4.3, line 2, read: S-purity

of d> k

in B ,

-^-^ ^

P. 32 Proof of Theorem 7.2, line 9, read: n(a*' + a') = a instead

of n(a'' + a) = a.

P, 35 line 12, read: since o<, A is an isomorphic image of A.

P. 36 line 10, read: r € Horn

( , r\) »

P. 36 line 14, read: Horn (B,o(A/^(^^p =/2.'Hom

©

P. 46 Definition 11.1, line 5, read: "gener-'.tor a" instead of

"generators a".

-2-" ..

P. 51 line 7, read: Therefore, 9 o( (l) = o( (l).

P, 79 bottom diagram, read: „

0 > A — ^ B

\ /

g \ ^/ h (m^n)

P. 87. Theorem 19.3 ( i ü ) , read: The exact seqnence

0 > c/j^ > G' ' > A > 0

splits.

P. 95 last line, read: n^N.

P» 95, add : tG denotes the maximal torsion submodule of the

module G.

CHAPTER I Introduction 1 1. Preliminaries 2 CHAPTER II S-divisible and S-pure subgroups,, 19

2 . S-divisible groups 19 3 . S-pure subgroups 22 4 . S-pure exact sequences 24

5. S-pure projective and S_pure injective f^roups 26

6. S-pure essential extensions 30 7 . S-pure extensions , 32

CHAPTER III D„-divisible modules and D^-pure submodules

over a principal ideal ring 37

8. D -divisible modules 37 9 . D_-pure submodules 41 10 . D„-pure exact sequences 43

11. Dg-pure projective and D -pure injective modules.. 46

12 . D -pure essential extensions 48 13. Modules of Do-pure extensions 50 CHAPTER IV Do-divisible modules and D^-pure submodules

o o

over a Dedekind ring 57 14 . Do-divisible modules 57 15 . Do-pure submodules 66 16. Do-pure exact sequences 69 17. Do-pure projective and D -pure injective modules.. 72

18. Do-pure essential extensions 81 19 . Modules of D^-pure extensions 85

References 93 Samenvatting 94 Table of notations 95

1 . CHAPTER I

Introduction

The concepts of divisible group and of pure subgroup are of great importance in the theory of abelian groups . I t has been found that every direct summand of a divisible group is divisible and every divisible subgroup is a direct summand of the containing group. The notion of pure subgroup is found intermediate between subgroup and direct summand. Hilton and Yahya [^] studying divisibility in

groups, generalized the notion of divisibility, In-stead of considering divisibility by natural numbers they defined (for a set S of prime numbers) S-divisibility. They stressed the simplicity of the concept of S-divisibility and studied S-extensions and S-injectivity in groups. In another paper R Q ] , Yahya defined the concept of S-purity and

studied S-pure extensions on the lines of pure extensions, an idea introduced by Harrison f^"] .

As the study of modules developed, Kaplansky in his book ^g'j

generalized the concepts of divisibility and purity to modules over a principal ideal ring and he found that many results which are true for groups are still true in case of modules over a principal ideal ring. In another paper [-rjl , Kaplansky studied the concepts of

divisibility and purity in modules over Dedekind rings and valuation rings. Finding the importance of the idea of the group of pure

extensions (introduced by Harrison [5]) in group theory, Nunke [n] studied the modules of extensions over a Dedekind ring and gave the idea of pure extensions in modules over a Dedekind ring.

From time to time the concepts of divisibility and purity were generalized for groups in one way or the other. The main object of this thesis is to generalize the concepts of divisibility and purity for groups to S-divisibility and S-purity respectively, where S is a set of prxmes, S the multiplicative closure of S. S defines a

dual ideal D in the lattice L ( Z ) of ideals of the ring of integers Z, S ^

generated by the elements in S . For modules introducing the concept of dual ideal D„ in the lattice L ( R ) of ideals of the ring R, we generalized the concepts to D -divisibility and D -purity. After dealing with the properties of D -divisibility and D -purity \<fe have developed the theory of Do- pure exact sequences, D -pure injective,

D -pure projective, D^,-pure essential extensions and modules of D -pure

S o o

extensions. Our generalization is such that when we consider the ring of operators as the ring of integers Z and S the set of all primes then the theory of S-divisibility and S-purity coincides with the ordinary concepts of divisibility and purity for groups;

on the other hand when we consider D to consist of all ideals in the lattice L ( R ) of ideals in the ring R, then the theory of

D -divisibility and D -purity coincides with that of divisibility and

D O

purity in modules over the ring R introduced by Kaplansky ['^J . First, we start with groups, i.e. modules over the ring Z of integers in Chapter II. ¥e introduce the concepts of S-divisible groups and S-pure subgroups and we study their properties in the subsequent paragraphs. Moreover, in Q 5, the concepts of S-pure projective and S-pure injective have been studied, after that the idea of S-pure essential extensions and groups of S-pure extensions have been dealt with in -S 6 and 5 7 .

In the next two chapters III and IV the above concepts of S-divisibility and S-purity for groups have been generalized to

D -divisibility and D -purity first for modules over a principal ideal ring and then for modules over a Dedekind ring. The ideas of D -pure projective, D^-pure injective and D -pure essential extensions have also been studied for modules over a principal ideal ring in oil and

b 12, and for modules over a Dedekind ring in 6 17 and 9 18. In the last, we have studied the modules of Do-pure extensions over a principal ideal ring in $ 13 and over a Dedekind ring in 5' 19, by means of

homological methods. 5 1. Preliminaries.

o

Some of the definitions and results \;rhich will be used frequently in course of our discussion are mentioned below, and unless otherwise stated they can be found in [ 4] or [10].

Groups: Groups in which every element has finite order are called torsion groups; those in which all elements j^ 0 are of infinite order are torsion free. By a p-group is meant a group in which the orders of the elements are powers of the same prime number p. If there exists an upper bound for the orders of the elements of a group G then G is

called a bounded group. Every bounded group is a direct sum of (bounded) cyclic groups. Every torsion group is a direct sum of

3.

p-groups for various primes p £ P. A group G is called divisible if

nG = G for all n £ N. If H is a divisible subgroup of G, then H is a

direct summand of G, Conversely, if H is a direct summand of every

group containing it then H is divisible. Any group can be embedded

in a minimal divisible group. Every group G has a unique decomposition

G = dG©R, where dG is maximal divisible subgroup of G. If dG = 0,

then G is said to be reduced.

A subgroup H of a group G is called a pure subgroup if the

equation nx = h € H ( n C N ) is solvable in H whenever it is solvable

in G; or equivalently, if nH = Hr^nG holds for all natural numbers n.

Every direct summand of a group is pure. If H is a subgroup of G such

that the factor group G/„ is torsion free, then H is pure in G,

In particular, tG is pure in G,

Rings: Let R be an integral domain. An ideal A of the ring R is

called a prime ideal of R if from ab € A it follows that atleast one

of a and b is in A; alternatively, A is a prime ideal if ab € A and

a ^A implies b € A. An ideal A ^ R of the ring R is called a

maximal ideal of R if there is no proper ideal B of R between

A and R, that is, if A r:l B r R implies A = B o r B = R. Every maximal

ideal of R is a prime ideal of R. An ideal A generated by one element

a in R is called a principal ideal, denoted by<a>; a ring R is

called a principal ideal ring if R is an integral domain in which

every ideal is a principal ideal. Then, of course every ideal of R

is finitely generated and so the ideals of R satisfy the ascending

chain condition. Also, a principal ideal ring is an unique

factoriza-tion domain, that is an integral domain in which every non-unit a ^ 0

can be factored as a product of primes and these primes are uniquely

determined upto unit factors. A Dedekind ring R is an integral domain

in xirhich every ideal ( ^ (O) , ^ R ) is uniquely a product of prime

ideals. Moreover, every prime ideal of a Dedekind ring is maximal and

every ideal is finitely generated. The sum A + B of two ideals A and

B of a ring is defined as:

A + B = j a + b | a £ A, b € B f

The product AB is defined as:

AB = < ,S a.b. a. e A, b. £ B f

( l=( 1 1 ' 1 ' 1 )

If A and B are two ideals in a Dedekind ring R and let A = P, 1... «Pi^Tc

and B = P-, 1....P, ic, m. , n. £ Z, where

{

P. | are distinct prime ideals

1 o ic 1 1 1

, „ K ^min (m, ,n.) ^ ^ -o r; r.™^x (™- >^-) A + B = n P. ^ i' 1'^ , A ^ B = n P. ^ i' i'

Let R be a commutative ring with identity, L(R) the set of all ideals in R ordered by inclusion. Since union and intersection are defined in L ( R ) , it is a lattice of the ideals in R. ¥e shall say that D is a dual ideal of L ( R ) if and only if

( i) D is a nonempty subset of L ( R ) ;

(ii) if A,, A„ belong to D then A /^ A_ £ D;

(iii) if A £ D then all the ideals B containing A belong to D. We now give some examples of dual ideals D of the lattice L ( R ) corresponding to a ring R:

(a).Let R = Z, the ring of integers and S a set of primes, S the multiplicative closure of S ( if S = ff, then S = {1|}. Now, we define D as the set of ideals < n > £L(Z) where n £ S , then D is

S o a dual ideal of L ( Z ) ; < n > C<iii>if and only if m|n; moreover, if S

is the set of all primes, then S = N , and D coincides with the lattice L ( Z ) of all ideals ( ^{o) ) of Z .

(_b) .If R is a principal ideal ring and S a set of primes of R, S is the set of all nonzero elements of R generated by the elements of S. Then we define Do as the set of all ideals <n> £ L ( R ) where n £ S and Do is a dual ideal of L ( R ) . Since, if <n>, <m> are two ideals in D then <n> r^ <m> £ Do, because, if <n> - <p^l....prk> and

<m> = <p-, 1... .pric> then < n > ^ < m > = <![ p. ^ 'i' i^ where each p. £ S. Hence < n > ^ < m > £Do. Also, if A £ Do and B is an ideal

1 ^ k k

containing A then B £ Eu . Since, if A = <p, l....p n> C B d <p > = I k k X n

maximal ideal .then p | p-,l....p n and hence any p. = p, that is p £ S, ^ £ Dg.

(c).If R is a Dedekind ring, S is a set of prime ideals of R and S is the multiplicative closure of S,then S*i ië a^dual ideal'Dg of

L ( R ) . Since, if A and '^ belong to D^, A = P^l pfk, B = P^l pfk /• \ ' D X iC X K then A r> B = è P™^^*^ ^i'^i^* £ D^, where P^ , , P, £ S. 'ïïow, if

•, , J X S -L -K

A £ D then to show that the ideal B containing A also belongs to D-If A = P-, 1.., ,P/k CP-jl....Prk = B, then since every prime ideal is maximal, by [ 2 1, P. = P. and n. = m..that is B £ Do.

'- -^ X J X J ' S Modules: Suppose R is a commutative ring with identity 1 £ R, then

an abelian group G will be called an R-module provided there is a product rg £ G corresponding to each r £ R and g £ G such that

r(gx +

=

+ rg2, r £

,

£

5*

(r + s)g = rg + sg

Ig = g , 1 £R.

Every abelian group is an R-module, where R is the ring Z of integers,

every vector space over a field K is a K-module. If R is a ring,

B is an ideal of R, then B is an R-module.

I f G i s a n R - m o d u l e , t h e n t h e s e t I = { r £ R | r g = 0 , V g £ G }

is an ideal ofR called the annihilator of G. If J is any ideal

contained in I, then for any element j £ J, we have jG = 0. Therefore,

every element of the coset r + J in R/ acts on G exactly as r does,

J

and we can consider G as an R/, module, by definition (r + j)g = rg.

If H is a subset of the R-module G, then H is called an

R-submodule ( or only submodule) of G if H is a subgroup of G such

that for every r £ R, ( rh | h £ H } = rH C H . If H is a submodule

'of G then the quotient group G/ is also an R-module with module

multiplication given by r( g + H) = rg + H, The R-module G/ is called

the quotient (factor) module of G by H. An R-module G is said to be

cyclic if G can be generated by a single element, i.e G = Rx for some

X £ G. If G and G are two R-modules then T) : G > G is an

R-homomorphism if

( i ) T] ( % + g j ) = Ti g-L + Ti g2 f o r g^ , g2 ^ ^» ( i i ) r ) ( a g ) = at] ( g ) f o r g ^ G , a £ R .

I

In the future, we say only: T) is a homomorphism of G into G . If the

kernel of the homomorphism is K and the image H, then K is a submodule

of G and G/ S H.

Let R be an integral domain. For any x in the R-module G the

set of all a £ R with ax = 0 forms an ideal in R called the order ideal

0(x) of X in R. The set of all elements in G with a nonzero order

ideal forms a submodule T of G called the torsion submodule; if

T = G we say G is torsion R-module, if T = 0, we say G is torsion free.

For an R-module G, the quotient module G/„ is torsion free. We may

observe that any cyclic R-module G = Rx is isomorphic to the R-module

R/^/ \; for, let G = Rx and 0(x) be the order ideal of x, the mapping

a > ax( a £ R ) is a homomorphism of R into G with kernel 0(x).

Hence G^ R/^^^y

An R-module F is called free, if there is a subset B of F with

the property that x £ F can be expressed uniquely in the form

x = S r.b. , 0 ^ r . £ R , b . £ B .

A free R-module is projective (see p.9 ) . Every R-module is isomorphic to a quotient of a free module. Since R is itself free R-module, any free R-module is any direct sum of isomorphic copies of the R-module R. A module is finitely generated if it is generated by a finite subset, that is, it is isomorphic to a quotient module of a finite direct sum R $& R ® • • • • ® R •

Kaplansky [ 7] introduced the concept of divisibility and purity in an R-module where R is an integral domain. The submodule A of an R-module G is said to be pure if rA = A'^ rG for every r £ R. If we denote G/. by B then the purity is equivalent to the following: for each element b £ B there is an element g £ G mapping onto b and having the sa-me order ideal as g. Again, this is equivalent to the relation lA = A ^ IG holding for every ideal I in R. If A is a submodule of an R-module G such that G/ is torsion free„then A is pure in G. In particular, tG is pure submodule of G. Any direct summand is pure. Purity is transitive property. If G is torsion free then the intersection of any set of pure submodules of G is again pure. Hence any subset of G is contained in a unique smallest pure submodule of G. An R-module G is divisible if rG = G for all nonzero r £ R. If G is divisible, then a submodule of G is divisible if and only if it is pure; also if G is any R-modulo then any divisible submodule of G is pure. A torsion free divisible R-module can be considered as a vector space over the quotient field K of R and conversely.

Categories and Functors: Let ra be a class of obj ects A, B , C , . . . . together with a class of mappings (called morphisms) a , p , y ,.... If to each pair ( A , B ) of objects there is associated a set hom(A, B ) of morphisms such that every a £ hom(A,B) has domain A and ( codomain) range B, and if for every a £ hom(A,B), B £hom(B,C) we have associated an element of hom(A,C) written as pa , then (e' is called a Category if the following two axioms a-ro satisfied:

(i) if a £ hom(A,B), p £ hom(?, ,C) and y. £ hom( C ,D) then

(YP)OC = Y(pa) '

(ii) for each object B there exists a morphism e £ hom(B,B) such that p Eg^ = p for all P £ hom(B,C), and e^a = a for all a £ hom(A,B).

Let iji and ^ be two categories, A covariant functor T on [a

to *~> is a function which 3-ssigns to each object C £ \o an object T ( C ) £ .y^) and to each morphismy : C > C of i;^' a morphism

a

11

7.

(i) T(l,)=l,(,^ , C € f p _

(ii) T ( P Y ) = T(p) T ( Y ) , PY defined in (^ (where p: C > C , C t {;J)

The covariant functor T is called additive if

( iii) T( a +Y ) = T( a) + T( Y) where a , Y : C i C .

A contravariant functor T on f^; to

JJ

is a function which assigns to

each G £ fp/ a T ( C ) in fö and to each morphism Y : C > C

morphism T ( Y ) : T ( C ) j. T ( C ) of £ ) satisfying:

(ii) T ( P Y ) = T ( Y ) T(P ) , PY defined in (:^ , p : C ^ c'

T is called additive contravariant functor if together with (i) and (ii)

(iii) T( a +Y ) = T(a ) + T(Y ) holds, where a: C ^ C .

Functor Hom: Let R be an integral domain; A and B be R-modules. The

set Homj^(A,B) = { f | f : A > B } of all R-homomorphisms f of A

into B is an R-module when rf : A > B is defined for r £ R and

f : A ^ B by (rf) (a) = r (fa) for all a £ A. That rf is still

an R-module homomorphism follows from the fact that

(rf)(sa) = r(f(sa)) = rs( fa)

:

:

= s[(rf)(a)]

using the commutativity of R. If R is a field, Hom_(A,B) is the vector

space of all R-homomorphisms of the vector space A into the vector

space B. If R = Z , then Homr7(A,B) is the abelian group of homomorphisms

of A into B. Also, if A = B, Homü(A,B) is

B.

ring under addition and

composition of homomorphisms, this ring is called the ring of

R-endomorphisms of A. If we fix the module A and p is a fixed

t

homomorphism of B into B . Each f : A > B determines a oomposite

pf : A >• B and p (f + g) = p f + pg. Hence the correspondence

f ^ pf is a homomorphism

p^ : Homj^(A,B)

h

Homj^(A,B')

of R-modules. It can be seen that Homp(A,B) is a covariant functor of B.

Again, if we fix the module B, then for a fixed module homomorphism

I t t t

a : A > A , each f : A > B determines a composite f a : A — > B

with (f + g ) a = : f

a +

g'a . Hence the correspondence

a '•

f ^ f a is a homomorphism

a"" : Hom^(A' ,B) > Homj^(A,B)

of modules. It can be observed that for a fixed module B, Homp(A,B) is

a contravariant functor of A. We say: Hom is a functor in two variables,

contravariant in the first and covariant in the second from the

Hom^(© A^,B) S n^ Hom^(A^,B) and also

Hom^(A, T T B . ) ^ n Hom„(A,B)

Exact sequences: A sequence of R-modules A. and R-homomorphisms a.

«1 as o£

A. ^ A, -^ A, ^ 0 ^ 1 2 is exact if Ima. = ker(a . ^) for i = 1, 2

1 ^ 1 + 1 ' ' , k-1.

• ^ . .(k>,2)

In particular, 0 ^ A monomorphism, while B

-^ B is exact if and only if a is a

P

epimorphism. The exactness of 0 ^ A

-^ 0 is exact if and only if p is an —V 0 is equivalent B

to the fact that a is an isomorphism. V/e call an exact sequence of the form

0 4 A a i-4

_{? c}

a short exact sequence; here ais an injection of A into B such that p is an epimorphism with I m a = ker(p). [observe that in this case A can be identified with the submodule Ima of B and C with the quotient module B/, ]. The exact sequence (l.l) is called pure-exact if a A is pure in B.

The diagram of R-modules and R-homomorphisms •^ A - 2 - ^ B —Ê-9 C •^ 0 0 •^ A a 4'E

X^5

P ¥ = Xp • S ^ C . In general, a diagram of R-modules and

R-homomorphisms is commutative if two paths along directed arrows from one module to another module yield the same composite homomorphism. The rive Lemma: Let a coimiutative diagram of R-moduler. with exact rovrs:

A 1 ai •^ A 2 «2 -> A •^ b 3 as -> A ₄ 04 •^ A

5

3"

( i) a-, , ttp, a-, ttq are isomorphisms, so is a^.

(ii) a is an epimorphism amd a , a are monomorphisms then a,, is a monomorphism.

(iii) a^ is a. monomorphism and a „ and a . are epimorphisms then a^ is an epimorphism,

9.

The 3x3 Lemma; If in the following commutative diagram of modul 0 0 0 es 0 -^ A. -> B^ ) C^ > 0 -^ A, -> B^ ) C^ ) 0 ^ A, 0 -> B 0 ^ C 4 0 0

all the three columns are exact and if the first two or the last two rows are exact, then the remaining row is also exact.

Projective and Injective 'lodules: A module P is called projective if given any epimorphism cT"; B ^ C, each homomorphism Y: P ^ C can be lifted to a homomorphism /3 : P > B such that '^A = Y . If R is an integral domain, and ö~ be such that ker(6") is pure submodule of B, then P is called pure projective. Every free module is

projective. An R-module P is projective if and only if it is a direct summand of a free R-module. A direct sum of R-modules is projective if and only if each summand is projective.

An R-module Q is said to be injective if given any monomorphism 'X from A into B, each homomorphism i.\ from A into Q can be extended to

/3 : B 't Q such that 169C = o< . Q is a pure injective R-module if the extension of '•< : A >• Q is alv;ays possible for a pure exact sequence 0 ^ A —^^^ B (where R is here an integral domain) . An R-module Q is injective if and only if Ext„(R/ ,Q) = 0 for every ideal L of R. A direct product of modules is injective if and only if each factor is

injective. An abelian group (Z-module) is injective if and only if it is divisible. Moreover, every R-module is a submodule of an injective R-module.

.2).

Extensions of *!odules Let A a^d C be modules over a commutative ring R with identity 1. An extension of A by C is a short exact sequence

E : 0 •> A - 1 ^ B - ^ - C 5> 0

of R-modules and. R-homomorphisms y.,^6"„ A morphism 4 : E ^ E of extensions is a triple i = (•=<.,A,Y) of module homomorphisms such that

the diagram

E : O ^ A > B > C > 0

» ^

E : 0 ^ A • B ^ C ^ 0

is commutative. In particular, taking A = A and C = C , two extensions E and E of A by C are equivalent (E B E ) if there is a morphism

(l.,f5,lo) : E > E . When this is the case, the five lemma shows that the middle homomorphism & is an isomorphism. In that case the extensions E and E of A by C are called equivalent. Hence equivalence of extensions is an equivalence relation. Extpj( C ,A) denotes the set of all equivalence classes of extensions of A by C.. The direct sum

0 > A ^ A 0 0 ) C > 0

is known as splitting extension of A by C. Any extension by a projective module P is splitting, i.e. Extp(P,A) = 0 for all A, Ext„( C,A) is a contravariant functor of C and covariant functor of A where A and C are fixed respectively.

Lemma 1.2 1. lOJ : If E is an extension of an R-module A by an R-module C and if / : C ^ C is a module homomorphism then there exists an extension E of A by C and a morphism

J = (l_/^,B>y) • E )• E. The pair (4 ,E ) is unique upto equivalence

of

'

Lemma 1.3 LlOj '• If E is an extension of an R-module A by an R-module C and if o(, : A > A is a module homomorphism then there

is an extension E of A by C and a morphism ^ = (»<. ,/'••,,l^) : E ) E . The pair ( j , E ) is unique upto equivalence.

If we consider any extension E : 0 } A ) B ) C ) 0 of an R-module A by an R-module C, then the classes of equivalent extensions form an R-modtile, vfhere R is a commutative ring with identity. We shall give a brief sketch of the proof:

To each c i C choose a representative u(c), that is an element u( c) in B with u(c) = G. ïor each r é R, the exactness of E gives

ru(c) - u(rc)^. X A ; similarly, e,d t C have u( c+d) - u( c) - u(d)fc.'!XA. Hence, there are elements f ( c,d) and g(r,c) in A with

u(c) + u(d) - f(c,d) + u( G+d) , c,d é C

ru( c ) = g(r,c)+u(r,c) , r t : R , c feC. The pair of functions (f,g) is called a factor system for the

1 1 .

f ; C X C ^ A and g : R x C > A satisfying the conditions stated on page 10. F_(C,A) is a group under termwise addition, that is, with (f^+fp) (c,d) = f,(c,d) + f„(c,d); and since R is commutative,

Frj(C,A) is an R-module. The factor system for a given extension E is not unique: For any different choise of representatives u (c) we must have u ( c) = 'Yh(c) + u( c) for some function h on C to A, One

calculates that

u'(c) + u'(d) = 7^[h(c) + h(d)-h(c+d)+f(c,d)j + u'(c,d) ru ( c) = 'y(..[rh(c) - h(rc) - g(r,c)l + u (re). The new factor system (f ,g ) for representatives u xs then gxven by the expressions in the brackets in these equations. This can be expressed in a different way also: To each function h on C to A there is an element ( <5-~,h, Q,h) <^ F „ ( C , A ) defined by

(£^h) (c,d) = h(c) + h(d) - h(c+d) (6 h) (r,c) = rh(c) - h(rc)

The factor system (f ,g ) for representatives u then has the form (^ >g ) - (f>g) + ('-VI-^>'-'TJ^) • Conversely any such function h can be

OK ^

used to change representatives in an extension. Thus, v/e set

T „ ( C , A ) the submoduie of all those pairs of the functions in

F „ ( C , A ) of the form (ooh,cC h ) ; the factor system (f,g) of E is

uniquely defined modul T„( C ,A) . Use the factor modtil P ( C ,A)/^ /_ «\

K K L^ (j ,A) ;

to each extension E assign the coset :1O(E) of any one of its

factor systems (f,g) in this module P-n(C,A)/„ / . N . ThenW(E) is uniquely determined by E , An equivalence of extensions map

representatives to representatives, hence the equivalent extensions have the same factor systems. It follows that 60 is a 1-1 mapping of the equivalent classes of extensions to the subset of the R-module F ( C , A ) / / X. It can be proved that Ext„(C,A) is a submodule of the

module V ^ ' ^ ) / T R ( C , A )

-Lemma 1.4 ^lOj . Let E : 0 > A ) B > C > 0 be an exact sequence of R-modulos. A homomorphism o( : A ) G ca.n be extended to a homomorphism B ^ G if and only if the extension °(E splits.

Since carrRnpondie^' to each fk. : A ^ G we. ;j,ot an ex:tenr;ion E, hence there ir, an_R-homomorphism

E*" : Hom^(A,G) > Extj^(C,G)

E is called the connecting homomorphism for the exact sequence E . Lemma 1.5 ClO]. Let E : 0 > A > B > C > 0 be an exact

sequence of R-modules. A homomorphism Y ; G )• C can be lifted to a

homomorphism Y : G ^ B if and only if the extension E Y splits.

Since corresponding to each / : G ^ C we get an extension E Y ,

hence there is an R-homomorphism

i^x. : Hom^(G,C) > E x t ( G , A ) .

If E : 0 > A ^ B > C ) 0 is an exact sequence of

R-F.odules, then the sequences

0 > Hora^(C,G) ^ Hom^(E,G) > Hom^(A,G)

- ^

Extp^C,G)

J-^-E^t^{B,G) ^ ^

Extj^(A,G)

and

0

y

Homj^(G,A) ^ Honj^(G,B) ^ Hom^(G,c)

^i-» Extp(G,A) Ji^l^ E x t J G , B ) _ ^ E x t J G , C )

are exact, for every R-module G (where R is a commutative ring with

identity). If R is a Dedekind ring then 0 can be added on the right

side of the sequences. If G.(i ;i l) is any family of R-modules, then

Extj^( C , IT G. ) .3^lTExtj^( C ,G. )

and ''' ^

-Ext^( © G, ,A) '^ I T Ext„( G_^ ,A) .

-"- L t T.

Also, if A is injective, then Ext_(C,A)=0

3)

Direct and Inverse Limits : Let R be a comautative ring with identity

and the modules are R-modules. Let -^A. ,• be a system of R-modules

indexed by a partially ordered set I, which is directed in the sense

that to i, j f. I, there is always an index k £ I such that i £, k and

j

C

k. Suppose that for every pair i, j C I with i ^ j , there is given

an R-ho .no morphism

IT'.

: A. > A.(i ^ j) subject to the conditions:

X X J

(i) 'n. is the identity map of A. for all i é I;

(ii) if i i j c k then r\^ r r j = T\^.

Then the system A^ = ^ A.(i é l ) ; T T . jis called a direct system.

We form the direct sum (J-) A. = A of R-modules in the direct system A

together with the submodule B generated by

a. - j y , ^ a . ( i . £ j )

The direct limit A ^ of the 'system _A is defined as the quotient module

A/g.

1 3 .

t h e d i a g r a m s

( i i j )

a r e a l l coiramitative; i f A = A A. ( i é: l ) ; TT"? )• , B = ) B.( i t l ) ; p •? I a r e two d i r e c t s y s t e m s of R-modules w i t h t h e same i n d e x I , t h e n by a n R-homomorphism cb : ^ > B i s meant a s y s t e m of R-homomorphism O = ) <p. : A. > ^ - ( i é l ) ) s u c h t h a t t h e d i a g r a m s A. "^ > A X * , B . X

9:

^h

are all commutative. Moreover, if A and B are two direct systems and

<h : A ^ B is an R-homomorphism, then there exists a unique R-homomorphism <b : A ^ B such that the diagrams

A.

Jli-^k

I

If A, B and C =i, C.(i é l) ; CT^ [are three direct systems, and if

S : A ^ B , -^jT: B ^ C are R-homomorphisms between them such that, for every i, the sequences

A. ^ B, - ^ 1 ^ C. are exact, then we say, the sequence

A ^ > B — i — > C is exact.

Lemma 1.6. If A, B and C are direct systems as above and if 0 ^ ^.—^A . — U B.^ii_> C, B.

X X

^ 0

is exact for every i é- I, then the induced sequence for the direct limits

0 -^A^- 4 B,

^ c

Suppose A.(i ^ l) is a system of R-modules indexed by a directed set I, and if for each pair i, j t I with i £ j there is given a homomorphism TT . : A. > A. (i i: j) such that

(i) i \ . is the identity map of A. for each i £ I, ( ii) for all i <. j -C k in I, we have TT •? TT ^ = T\ ^.

Then the system S = ^ A.(i £ l ) ; TT . i is called an inverse system.

j_ 1 ^ ' 1 J '

The inverse limit of this system A"*^ = lim^ A. is defined to consist

J' 4 1 X

of all vectors a = (

"1^ ,.,.... ) in the direct product

A = . 1 ! A. for which TT . a. = a. (i ^ j ) holds. A is a submodule of A. Moreover, there exist R-homomorphisms

that the diagTams

A ^ A. ( i fc l ) s u c h

1 ^ ^

A

a r e a l l c o i m m i t a t i v e .

E x a m p l e . L e t A. ( i £ l ) be submodules of a n R-module A and l e t \ A . . D e f i n e ± c en- f o r a l l i è I and i ^ j f o r i , j é I U < cc>\ t o mean t h a t A. 'C, A. and l e t T] • be t h e i n j e c t i o n map A. > A. .

J - x 'x "" - ^ j ' x Then A = 1 A. (i e I vj i col , TT • } is an inverse system with inverse limit A . In fact, A = lim A consist of all vectors in 11 A. where coordinates are the same elements of A^.. This example shows that intersections may bo regarded as an inverse limit.

The notion of R-homomorphisms for inverse systems can be

defined analogously to direct systems. Let A = i^ A. (i c l) ; T\ . l , B = •! B. (i è l) ; f . (- be inverse systems of R-modules, indexed by the same set I. An R-homomorphism i-p : A ) B is a set of R-homomorphism

A.

Ii

A

B.

ƒ,,

f.

4B.

be commutative.

If ^ : A > B be an R-homomorphism between the inverse systems A and B, then there exists a unique R-homomorphism ip : A ^ B such that for every i *c I the diagram

A"

r

ƒ-.

i.

15.

is commutative.

Lemma 1.7. Let A = i A^ (i ^ l ) ; M M , B = ^ B^ (i £ l) ; f ^ and C = 1 C. ( i t l);ii~. \ be inverse systems with the same index set I, and ^ : A > B , '']r : B > C be R-homomorphisms such that the sequence

0 ^ A > B ^ C

is exact, then for the inverse limits, we have the exact sequence 0 > A*" ) B* ^ C*

Tensor and Torsion Products: Let R be a commutative ring with identity. The Tensor product A ® B of two R-modules A and B is the R-module generated by the set of all pairs (a (i)b),a(.A, b ^ B subject to the relations

(a-.+ap) @ b = a, ® b + a„ cè" b , a, , a„ t A, b fe B a ® (b-j^+b^) = a igi b-|^ + a (X) b^ , b-^^, b^ é B

(ra) ® b = r(a -g) b) = a QÖ rb , r é R, a € A, b e. B. This statement describes A .(X) B as a factor module F ( A , B ) / O / . -O\

where F ( A , B ) is the free module generated by the set of pairs (a,b) and S ( A , B ) is the submodule of F ( A , B ) generated by all the elements of the form

(a-j^+a^jb) - (a^,b) - (a^ ,b) ; (a,b-^+b2) - (a,b^) - (a,b2) (ra,b) - r(a,b) ; (a,rb) - r(a,b)

Also, A ( g ) B S . B ( g ) A ; R ® A ' ^ A a n d B ( g ) R = - B

defined by natural isomorphisms r ® a ^ ra and b ® r ^ rb respectively, where r e. R, a & A , b é B. Moreover, if

0 » A V B > C y 0

is an exact sequence of R-modules, then the sequence G (X) A > G ® B ^ G ® C ^ 0 is exact.

The tensor product is additive functor covariant in both

variables on the category of R-modules to the category of R-modules. Let R be any ring, a chain complex K of R-modules is a family

^ K ,"^ I of R-modules K and R-module homomorphisms 'è' : K ^ K -, , l _ n ' n ] n ^ n n " n-1' defineAfor all integers n, - C/D Cn^cO', and such that"& "B , = 0,

° ' ' n n+1 ' that is, KerC^ ) "D ImT^i . A complex K thus appears as a doubly

infinite sequence K : i K_2 ^ K_^ < K Q i K^ i K^ <-with each composite map zero. If K and K are complexes, a

chain transformation f K

K ^ K one for each n, such that 'J f "" •" ' n n

-^ K is a family of module homomorphisms n n n

Hence a commutative diagram

f , o for all n. n-1 n K K •s.. K ^ 4-n-1 K ^lii: K , ^ n n+1 n-1 "^^ V n "n+1

Vl^

K ' fillip K ' , -^

exists. A complex K is positive if K = 0 for n < 0 and is negative if K = 0 for n y 0. Each module A may be regarded as a "trivail" positive complex, with A_ = A, A = 0 for n 7^ 0, and ^^ = 0.

Now let R be a conmutative ring with identity 1. We consider the chain complexe L of length n( ^ O ) ,

^ 0 ^ ^1 ^ ^-^ L^ 4r

1)„

L 1 ^

n-1 n where each L, is a finitely generated free R-module. The dual L ^ Hom^ (L,R) that means L. = Hom^ (L. , R ) , i = 0, 1, 2, .... can also be regarded as a chain complex L

L 4 L -,

<-n <-n-1

L o ^ n-2

L ^ ^

f.

L*' = Hom^ (LJ^,R) = Hom^ ( © R^ ,R) '^ITHom (R. ,R) r-- ® Rj_ 5 and 4 L, -, is defined from 0

k+1 k+1 -> L, -'k "^ "k+1 -^-' ---^--^ ---^- 'k+i • ^k+i ^ ^k'

If G is an R-module regarded as a trivial chain complex, a chain -^ G is a module homomorphism/^o : Lo ^ G transformation/M. : L

with/^^-^ 0

module homomorphism/lA, . ^^

-^ G, while a chain transformation ^•^ : L > C is a module hononorphism with uo = 0 , where C is an R-module. For

•^ n - 1 '

given R-modules G and C we ta»ke elements of Tor ( G , C ) all triples t = (yt^,L ,V),yU,: L > G, V -^ C , where L has length n and n u , vare chain transformations defined as above. If L is a second

such complex and ƒ : L ^ L is a chain transformation, then gQ is the dual f^: L ^ L . Givenyi^-' : L > G and V : L * > C, we suppose that

i/f,^r^) = {/^,L ,vf) (1.8)

These maps may be shovm by a pair of commutative diagrams G f-G £-^ 0 £-^ ^ 0 ^ ^ 1 ^

f,

»

17.

Therefore, two triples in Tor are equal if the second is obtained

' n

from the first by a finite succession of applications of rule (l.8),

This describes Tor as a set. This set is a functor. For maps

Y|: G > G' , Y: C ^ c' the rules

t^(

j\^,L,V)

= (>]/^,L,i;) and

•i^{^,L, V

)

= {y^L,i)J)

.... (l,9)

preserve the equality (l.8) and make Tor a covariant bifunctor.

Two triples t, and t„ in Tor (G,C) have a direct sum the triple

in Tor^(G ® G,G © C) . If t-^ =

t^

and t2 = t^ according to (l.8)

then t-, ® tp = t-, @ tp, IfCOp is the automorphism of G © G given

^y^(gx,g2^ " (gj'gj ^^^en

{t^($>t^) =

(cOp)^. ( t 2 ® t^) as can

be found by applying (l.8) with

P :

L ^ L > L 0 L the map

interchanging the summands.

If we define the addition of two triples t-, and tp by

*1 +

\

= V G ^ y c -(^^l©*2^ ^ Tor^(G,C) (l.lO)

with V : G (J) G > G and V : C ® G ^ C defined by

IT 0

%(gl>g2) = gi + g2 ^^^ Vc(°i'°2'^ "" °1 •*• °2 respectively. Then

Tor (G,c) is an abelian group. Moreover, Tor (G,C) is an R-module

when R is a commutative ring with identity.

>Jhen n = 0, TorQ( G ,C) ^-G (2>C . Again, for E £ Ext( C ,A) and

t £ Tor

the product Et is well defined element of Tor

which satisfies o( (Et) = (o(,E)t, ( E ^ )t' = E(Y^t) , E(r)t)

='^J

(Et)

where o(: A ^ A, Y : c' > 0,^^:0 > G* and t *£ Tor^( G,c') ,

Given E and G, a map

E^ : Tor ( G , C ) ^ Tor T(G,A)

^ rv- ' ^ ' n-1^ ' '

is defined as E ^ t = Et; therefore, the long sequence

... ^ TorjG,A) >Tor^(G,F,) ^ Tor^(G,C) ^ Tor^_^(G,A)

^ Tor^_^(G,B) > Tor,^_^(G,C) > ...

is exact, v;hen

E : 0 ^ A > B ^ C > 0

is an exact sequence. If P is projective, then Tor (P,C)

-

0 for n ^ 0 ,

VJhen R is a Dedekind ring, Tor (G,C) = 0 for n )> 1, hence the long

sequence reduces to the exact sequence

0 > Tor^(G,A) ) Tor^(G,B) > Tor^(G,c)

_Li^> G (2) A ^ G ® B > G ® C ^ 0

A l s o , we h a v e L l ] >

E x t ( A , E x t ( B , C ) ) ' ^ Ext ( Tor^( A , B ) ,C) f o r a n y R-modules A,B and C.

19. CHAPTER II

S-DIVISIBLE GROUPS AIMD S-PURE SUBGROUPS

In this chapter, it is our aim to push further the theory of S-divisible groups and S-pure subgroups, the ideas introduced by Hilton and Yahya \_6]. Also, in ü 4 and 0 7 , we study the S-pure exact

sequences and the group of S-pure extensions. Before starting with the main theorv, we shall give some definitions which will be used

throughout this chapter. It is understood that where no proofs are given, they can be found in the literature mentioned.

Suppose S is a set of integral primes, S* the multiplicative closure of S; if S = 0, then S"^ = ^ 1^. Then

( i) an S-torsion group G is a group with 0( g) £ S*^ for all g ^ G.

(ii) a group G is S-torsion free, if GLp] = 0 for all p f_ S. In fj,eneral, the set of elements of a group G of order belonging to S' form a subgroup T of G and G/ is S-torsion free.

(iii) a rvovL^ G is S-bounded if nG = 0 for some n £ S '.

(iv) a grouxD G is S-cyclic if it is cyclic and its order belongs o ^

to S or is infinite.

O 2. S-d,ivisible groups

Definition 2.1. An S-divisible group G is a group with the

property nG = G for all n é S . By an S-divisible subgroup we mean a subgroup which is S-divisible as a group. A group G is S-reduced if G contains no S-divisible subgroup 7^ 0.

An epimorphic image of an S-divisible group is S-divisible. A direct sum (restricted or unrestricted) of groups is S-divisible if and only if each component is S-divisible. If (G.V. ^ is a set of S-divisible subgroups of G, then l_j G. is S-divisible. Every group therefore contains a maximal S-divisible subgroup.

Suppose G is an S-divisible torsion group and G = © G , the representation of G as direct sum of its primary components ( G jl O) ,

Then the structure of different components is as follows: (a) G can be any p-group, if p ^ S.

(b) G is a direct sum of quasicyclic groups Z(p'-'), if p £ S.

IT

If G is an S-divisible torsion free group of rank 1, then G can be characterized by a height ( k, ,kp,....,k. ,....) where k. are

S-divisible groups of rationals, there is a minimal S-divisible group Q ( S ) , characterized by the height ( k, ,kp,....,k. , . .,.) with k. = co if p. t. S and othervfise k. = 0. Q(S) is the group of rationals -,

1 ;, J n n € S * (admitting 1 <c S ) .

Theorem 2.2 [61. A torsion free group G of rank r i_s S-divisible if and only if Q(S) C G Q Q .(denoting by A the direct product of r copies of A ) ,

Definition 2.3. An S-extension G of a group A is a group G containing A such that G/. is an S-torsion group; we also say that 0 ^ A ^ G is an S-exact_row.

Definition 2.4. A group G is S-injective if for any S~exact row 0 > A > Bj any homomorphism ^]: A > G can be extended to a

t

homomorphism ">'") : B ^ G; that means that the following diagram is conmiutative:

0 ^ A ^ B

G

' ' I

Theorem 2.5. Every group G can be embedded in an S-divisible S-extension of G.

Proof. Since any free group F = (^ '^^•^ oan be embedded in S-divisible group 0 Q^^'*(S) and G^i: F/^^, we have G C @ Q^^\s)/^^ this proves that G is a subgroup of an S-divisible group, using the fact that an epimorphic image of an S-divisible group is 3-divisible, Moreover, we have

©Q(i)(s)A/ - ® Q ^ i \ s ) 4 / ^ ®Q^^\s)A^ (i)^ (B@ z(i\p),

S-extension G of G.

Theorem 2.6. [^63, A group G is S-divisible if and only if G is S-injective.

Theorem 2.7. If a group G is an S-extension of an S-divisible subgroup A, then A is a direct summand of G, G = A (T^ B; moreover, if C is a subgroup of G such that A C) G = 0, then B can be chosen in such

21.

a way that C c. B.

Proof. In the diagram the row

0 ^ A .^i-^ G is S-exact and the 0 ^ A ^f^ G

identity map I. can be extended i

.

^

/J '-o'

such that ••>') o< = I. . Since A (~N C = 0, ''' i

- A' we have A © C = G and the

sequence 0 > A Q C ^ G

is S-exact, since G/ „ is a homomorphic image of G/., Defining

')'^: A (£) C ^ A by'>i(a+c) = a, we can extend T) to a homomorpliism -,y: G ^ A and C satisfying the property C C Ker ('>!')> '*7 defines on A the identity I,. Therefore, we have the decomposition

G = A ® B, C C B.

Since the union of S-divisible subgroups of a group G is S-divisible, we see that any group G contains a unique maximal S-divisible subgroup D. If therefore, G is an S-extension of D, we

know that D is a direct summand of G, G = D (j3 B. Now, E cannot contain any S_divisible subgroup C, otherwise, D ("^ C would become a proper S-divisible extension of D. Hence, we have G = D i;^ B where B is S-reduced. B is uniquely determined upto isomorphism.

Theorem 2.8. If the subgroup A of a group G i^ S-divisible and G/A is. S-divisible, then G is itself S-divisible.

Proof. To solve the equation nx = g (g -iL. G, n ^ S ) , we solve nx = g ( $2 G/^) ; since G/. is S-divisible, nh = g ( h £ G/. ) , then nh - g = a é A or nh - g = na (since A is S-divisible), hence n ( h-a ) = g. Therefore, G is S-divisible.

Theorem 2.9. Let G =: I I A (n=l,2, ) , A = (^, A , then G/. is S-divisible if and only if for every prime p ^ S, pA = A holds for almost all n.

Proof. If G/ is S-divisible, then n(x+A) = g + A is solvable for all n G. S " in G/ ; if x = (x-, ,X2 , . . . . ) , g = ( g-, ,gp , •. • .) then this condition means that ( nx,-g, ,nXp-gp,....) ^ A o r n x . - g . = 0 for almost all i and this m^ans pA. = A. for all p £ S and for almost all i

• ^ 1 1 ^

Conversely, if pA. = A. for all p € S and almost all i, then

n(x+A) = g + A or (nx-, -g-, , . . . . ,nx. -g. , . ...) ^. A can be solved. Theorem 2.10. Suppose A i^ S-torsion free, then its m;a.xii7ial

O 1

(i.e. ' 1 nA = Ao) if A is an S-extension of D ( A ) .

rv. t c »

Proof. Since the union of S-divisible subgroup of A is S-divisible, A contains a maximal S-divisible subgroup D(A). First, we have

D ( A ) (C ( \nA, since d é. D ( A ) implies d <i n/v for all n ^ S .

Conversely, A C . D ( A ) , A is an S-extension of D ( A ) , therefore, we

have by Theorem 2.7, A = D ( A ) Q) R , v/here R is S-reduced. Take a £ A , then a = d + r (d"£ D ( A ) , r € R) . Now, a = na , since

1 * "

a ^ nA, n € S ; a is unique, since A is S-torsion free. Hence, we ti ti II II X II X

have a = d + r , with nd = d , nr = r , but then R should be S-divisible and R • C - D ( A ) , therefore, we have A C D ( A ) .

Theorem 2 .11. A group A _is S-rcduced if and only if Q ( S ) has no non-trivial homomorphism into A.

Proof. If A is S-reduced, then Hom(Q(s),A) must be zero.

Conversely, if Hom(Q(s),A) = 0, then A cannot contain an S-divisible subgroup [^ O) . For, if 0 7^ A C A was S_divisible, then S-exactness of the sequence

0 9 Z ^ Q(s)

implies that any homomorphism 'H : Z ^ A with ^ ( l ) = ^r\i^ O) can be extended to a non-trivial homomorphism "."j : Q(S) ^ A (a t-, Ao) .

p 3. S-pure subgroups o

The property of a subgroup A of a group G being S-pu.re in G can be expressed by ni'V = A H nG for all n £ S ; or equivalently, v/henever v the equation n x = a f c . A , n é S is solvable in G is solvable in A, The usual properties for pure subgroups are still true for S_pure

subgroups. To familarize oiirselves with the concept of S-pure subgroups, we mention some ne\-i results:

If G is S-torsion free, then the intersection ' 'A^ of any set of S-pure subgroups A^^ is S-pure. Therefore, in S-torsion free group,. G there is a least S-pure subgroup K , generated by a given subset K of G. Ko consists of all those elements g € G which have an integral multiple n g t < K > , n t S . If the factor group G/. is S-torsion free then A is an S-pure su'^group of G. Therefore, the extension G ( A , B )

of a group A by an S-torsion free group B are S-pure extensions, i.e. A is S-pure subgroup of any extension G of A by B. We now state

23.

Lemma 3.1- If A is an S-pure subgroup of a group G and B/. is S-pure in G/, then B is S-puro in G.

Proof. Let ng = b Ó, B, g ^ G, n € S , then n(g+A) = b + A and by hypothesis there is some b é_ B such that n( b +A) = b + A or nb = b + a(a é A) and this implies nb - ng = a, by -J-purity of A in G, we have nb - ng = na (a é A) and n(b — a } = ng = b, where b - a C- B.

Lemma 3.2. The S-purity of a subgroup A of a group G is equivalent to the possibility of selecting in every coset g with 0(g) é. S* an element g t G with 0( g) - 0(g).

Proof. If 0( g) =<:^, then any choise of g mapping on g is of infinite order. Now suppose, o(g) = n é S , let any h £ G is mapped on g, then nh é. A and there exists an element a £ A such that nh = na. Let g = h - a, then n(h-a) = 0, and 0(g) = n £ S ,

Lemma 3.3. If A is an S-pure subgroup of an abelian group G such that G / is a direct sum of cyclic groups Z(n.) , n. è. S , eventually n. =cOf then A is a direct summand of G.

Proof, It is similar to the usual proof for pure subgroups. From this lemma, wo can conclude that the extensions G ( A , B )

with A an S-pure subgroup of G and B a direct sum of cyclic groups Z(n.), n. é S , eventually n. =co, are splitting.

Lemma 3.4. Let A be an S-pure subgroup of a group G such that nA = 0, n £ S'', then ^,A,nG"/ / „ is S-pure in G/ ^.

Proof. Suppose my = x , m t S j X Q. i(A,nG/ / p , y £ G / ^ , then we prove that x is an m-mulpiple in •CA,nG~)-/ . Let us take

representatives of x and y in G. The representative of x may be chosen in A, for a, + ng, = ^p + ngp (mod nG) only if a., = ap. For, if

a, - a„ = a t nG, then a = ng and by hypothesis a = na , n «i S ,

1 ^ t

therefore na = 0, since nA = 0, a = 0. Now suppose a is the represen-tative o.f X, t the represenrepresen-tative of y. Then a - mt = nz <ir. nG. Suppose r = g.C'.d(m,n), m = r m ^ , n = r n , (m, ,n-^) = 1. Since m, n é' S ,

their fa.ctors r, m, , n-, ^ S^ and we haveO\m-, +/3in = 1, and

a - rm-, t + rn, z = r (m-,t+n-,z); therefore, a can be written as a = ra-, . Since A is S-pure in G, r £ S , a = ra, = r (o<,m,+i^-^) a^= m-xa, + n^a, = m-xa, for na, = 0; take mod nG, we sec that x is an m-multiple in

Lemma 3.5 [_ 8 J , Let S and T be subgroups of G with S H T = 0, and suppose that <^S,T~'/ /„ is a direct summand of G/ . Then S is a direct summand of G.

Theorem 3,6. Let A be an S-pure subgroup of G such that A is S-bounded (i.e. there is an n t S such that nA=0) then A is a direct summand of G.

Proof. Suppose nA =; 0, n ^ S , then by Lemma 3.4, ^^AjnG") / „

is S-pure in G/ . Now G/ ^ is S-bounded and therefore all its . homomorphic images are S-bounded, and hence all the homomorphic images of G / „ are direct sums of cyclic groups Z(n.), n. è S . We have therefore, <CA,nG^ / „ is a direct summand. of G / ,,; furthermore,

' ' ' ' nG 'nG' ._ ' A O nG = nA = 0, hence, by Lemma 3.5, A is a direct summand of G,

Since the S-torsion subgroup of a group G is always E-pure, we see that the S-torsion subgroup is a, direct summand if it is of S-bounded order. Moreover, a finite S-pure subgroup A of order n <i. S is a direct summand.

Leiama 3.7. If A is an S-pure subgroup of a group G and T a subgoup of A, then A / _ i^ S-pure in G/ .

Proof. It is similar to the usual proof for pure subgoups.

Theorem 3.8. A subgoup A of a group G i_s S-pure in G if and only if for all n é S , A/ . is a direct summand of G/

Proof. If A is S-pure in G, then it follows from Lemma 3.7 that A / A is S-pure in G/ ., and since A/ is S-bounded, it is a direct

summand of G/ The converse can easily be checked. ,

P 4 . S-pure exact sequences A short exact sequence

D i

•''-0 ^ A _ll-> B — i ^ C )• •''-0 (4.1) is said to be S-pure exact if Im --•'. is an S-puro subgoup of B. It is

pure exact if S consists of all the primes and p-pure exact if S consists of only one prime number p.

Theorem 4.2. Suppose (4.l) is an exact sequence. Then each of the following conditions is equivalent to the S-pure exactness of (4.l):

(a) 0 ^ nA "^ > nB —^-> nC ^ 0 is exact for all n t S"^;

25. (c) O (d) O ->A/ °<.

nA

^^/nC -^ O is exact for all n £ S ; > O is exact for all n ^ ^

' A n •^ 'B n ^ ^C n

Proof, l'he composition of c<, and A are O throughout. If (4.l) is exact, then o( is a monomorphism and i'bxs epimorphism. The kernel of & in (a) is^xA O nB v/hich isü<(nA) if and only if (4.1) is S-pure exact.

In ( b ) , cX is a monomorphism; A : BfnT ^ C[n],is an epimorphism if and only if every element of order n of C is the image of an

element of B of order n (n t S ) . This is, by Lemma 3.2 is equivalent to the S-pure exactness of (4.I).

(c) Consider the following commutative dia,gra-m:

0 ^ nA -^. A -> nB nC -^ 0

^ c

Since the first two rows are exact, by 3x3 Lemma 0 — ^ A / ^.B/

_-^c/

is exact for n é S . I t can be c''^ecked easily that the exactness of above sequence implies that o^ A is S-pure in B.

(d) The exactness of (4.l) and (b), and using 3x3 Lemma we get the exact sequence

S —^

% [ n ] - ^

V B M — ^

'/C[nl—^ °

Since A/»j- _.= nA, B/ j- -, i; nB, C/Qr-.^nC, the above sequence reduces to 0 ^ nA ^ nB > nC > 0

But the exactness of this sequence implies (by (a)) thatc<A is S-pure in B.

S-pure exact. In view of Theorem 3.6, it is sufficient to show that the sequence (b) is S-pure exact. This follows from the fact that for any element c £ Gfnl , v/e have nc = 0 (n C S ) . Suppose 0( c) = n , n I h, then n c = 0 and n b = a, hence n a = a and n (b-a ) = 0; and b - a must have the order n , this proves the S-purity of A\^rA in B^n^ .

Theorem 4.3. Suppose A = ( A ^ , i £ I, T T ^ ] , B = 1 B^, i £ I, f ^ I and C =:-sC., i ^ 1 ,<S. i are direct system of groups and 4^ : A _{7 _ >} TI'": B ^ C are homomorphisms betvreon them such that for all i é I,

0 > A._Z±9, B.—il^ C. > 0

1 1 ^ 1

is S-purc exact, then the induced sequence between the direct limits

0 > A ^ - 1 ^ H^^^ C.^ > 0 is likewise S-pure exact.

Proof. Since the direct limits of exa-ct sequences arc exact (Lemma 1.6), v;e have only to prove the S-purity of 4~ A in B , Let a é A and b é. B satisfying nb = $> a for some n & S*^. Then there are i é I and a. C A. , b. c B. such that a = TT.a. , b = f-b. , because of

l l ' l ^ X X X ' ' X X '

t h e c a n o n i c a l ma-ps TT- '• A. > A and p. : B. ^ B . By t h e

'X X ^ >*. J i 1 '' ^ d

commutativity of the diagrams (p.l3), we have, nb = p.(nb.) = 4> a = - X X x-^

S~\\.B.. - ^.S.a., whence P. (nb. - ^.a.) - 0, and then there is a j ^ i such that P^(nb. _ ^ . a . ) = 0. Therefore, P'?(nb.) = P'?<J.a. =

.TT-^a. ; and since p9b. (E B.,lT'?a. £ A. and ] x x ' J X X j'^xx'^- 3

_±L>B _ Ï L

0 > A . —3-i-> B . -JJ-y c . > 0 <] Ü J

is S-pure exact, we conclude that there exists some g. g^ A.

satisfying-^^(ng^ '^'fjTTi^i» ^-PP^Yi^S fj ^'^^ since p^ ^^ ^ f '^

^ y

we obtain <fc'(ng) =<i'a,where g =Tr-g- é. A . This proves that the direct limits of S-pure exact sequence is S-pure exact.

Q 5. S-pure projective and S-pure injective groups

In the follov/ing vre shall describe the S_pure projective and S-pure injective properties of groups. Most of the results of this section are already knovrnf^n . Since this theory is quite important for its further generalization to modules, we are dealing it in a bit

27,

detail. Before starting the main theory, we shall give some definitions, Definition 5 .1. By a group of type p , vfhere p is a prime, we

mean the group Z(p ) , n^iyz>. We observe that a group A is cyclic with generator a if and only if, given any homomorphism f into A, a £ Imf implies that f is an epimorphism. Dually, a group A is of type p with generator a for its cyclic subgoup of prime order if and only if, given any homomorphism f defined on A, f (a) / 0 implies that f is a monomorphism. Also, for any group A and any a t-. A, there is a cyclic subgroup of A generated by a, and dually, if a / 0, there

n

is a homomorphism f defined on A v/hose image is of type p with cyclic subgroup of prime order generated by f(a); f may be defined as the natural homomorphism from A onto A/„, where C is a subgroup of A maximal with respect to the property of not containing a. We shall call a group S-cocyclic if it is of type Z(p^-'') , p is

arbitrary, or of type 2(p ) , when n is finite and p ^, S,

Definition 5.2. A '-':roup G is said to be S-pure projective if for any homomorphism f from B onto C such that Ker(f) is 3-pure in B, if g is a homomorphism from G into C, then there exists a homomorphism h from G into B such that g = fh. In other words, in any S-pure exact sequence ( 5.3) .G h / > f g ^ A ^ B - ^ C ->0 (5.3) if g is a homomorphism from G: .into C then if there exists a

homomorphism h from G into B such that the diagram is coiranutative, then the group G is called S-pure projective.

If S is contained in S then any S-pure projective group is also S -pure projective. If S is empty then S-pure projective groups are just projective groups. Also, any isomorphic image of an 3-pure projective group is S_pure projective and a direct sum of groups is S-pure projective if and only if all its summands are S-pure projective.

Lemma 5 .4 [^9"]. If f is a homomorphism from B onto A, then Ker( f) is S-pure subgroup of B if and only if for any homomorphism g from an S-cyclic group G in A there exists a homomorphism h from C into B such that g = fh, i.e.if and only if the diagram

.C

^ / g

/f ^

B -^-^ A ^ O is commutative for any S-cyclic group C.

Theorem 5 -5 [91 • If A is any group then there exists a

homomorphism f from a direct sum B £f S-cyclic groups onto A. such that Ker(f) i^ S-pure in B. In other words, the sequence

0 ^ Ker(f) > B -L-^ A > 0

is S-pure exact when B is a direct sum of S-cyclic groups.

Theorem 5 .6. The follovring conditions on a group A are equivalent: (i) A ±s_ S-pure projective.

(ii) If f is a homomorphism from B onto A and if Ker(f) is S-pure in B, then there exists a homomorphism g from A into B such that fg is the identity on A (i.e. Ker( f) is a direct suimnand of B ) .

(iii) A is a direct summand of a direct sum of S-cyclic groups. Proof. That (i) iiuplies (ii) is obvious. Assume that (ii) holds. By Theorem 5-5, there exists a homo-iorphism f from a direct sum of S-cyclic groups B onto A such that Ker(f) is S-pure in B and by (ii) A is a direct summand of B. Finally, suppose (iii) holds and we prove that A is S-pure projective. Since a direct STUIL of groups is S-pure projective if and only if each summand is S-pure projective, hence we may suppose that A is S-cyclic and then the condition of Lemma 5.4 is satisfied and hence A is S-pure projective.

It is wellknovm that any subgoup of a direct sum of cyclic groups is again a direct sum of cyclic groups. If one looks at the proof of the above theorem, one can easily find that any subgroup of a direct sum of S-cyclic groups is again a direct SUJU of S-cyclic groups, hence a group is S-pure projective if and only if it is a direct

sum of S-cyclic groups.

Lemma 5.7 r^l. If f is an injection of A into B, then A is S-pure in B if and only if for any homomorphism g from A into an S-cocyclic group C, there exists a homomorphism h from B into C such that g = hf (i.e. h is an extension of g ) .

Definition 5-8. A group G will be called S-pure injective if whenever A is S-pure in B and f is a homomorphism from A into G,

29.

then f is extendable to a homomorphism g from B into G, In othen-zords, G is S-pure injective if for any S-pure exact sequence (5.9) and any

0 ^ A - ^ B (5.9)

G"

homomorphism f from A into G, there exists a homomorphism g : B ^ G such that the diagram is commutative.

If S is empty then S-pure injective groups are just injective

t '

groups; also, if S C S then S-pure injective group is also S-pure injective. Furthermore, any isomorphic image of S-pure injective group is also S-pure injective and a direct product of groups is S-pure injective if and only if each of its factors are S-pure injective.

Theorem 5.10 If A is any group then there exists a direct product B of S-cocyclic groups such that A is S-pure in B,

Proof. Let -, f.(i é l) v be the set of all natural homomorphisms defined on A whose kernels are subgoups of A v/hich are maximal with respect to the property of not containing some non-zero elements of A, Then, by the definition, f.(A) is of type p.* . If p. is in S, set B. = f.(A) and if p. d S, set B. to be a group of type Z(p'^) and consider f. as a homomorphism from A into B.. Then, if B is a direct

1 •" 1 '

product of the B. , there exists a homomorphism f : A 5; B such that for each if- I, f. =g.f, vrhere g. is the canonical projection of B onto f.(A). Since the intersection of all the kernels f. is zero, f is an isomorphism, and by Lerama 5.7, f(A) is S-pure in B.

Theorem 5 - H . The follovring conditions on a group A are equivalent: (i) A i^ S-pure injective.

(ii) A is a direct summand of every group of which it is an S-pure subgroup.

(iii) A is a direct summand of a direct product of S-cpcyclic groups Proof. That (i) implies (ii) is obvious. Suppose (ii) holds. By

Theorem 5.10, there exists a direct product B of S-cocyclic groups such that A is S-pure in B and by (ii) A is a direct summand of B. Finally, suppose that (iii) holds and let us prove that A is S-pure injective. Since a direct product of groups is S-pure injective if and only if each ;factor is S-pure injective, we observe that we may

assume that A is S-cocyclic, and then by Lemma 5-7 the result follows. 0 6. S-pure essential extensions

The notion of S-pure essential extension is introduced by Maranda )_$)] where he develops its theory in full detail. In the

folloviing we shall describe it briefly and in the subsequent chapters this will be generalized for modules.

If A is a subgoup of B then we will denote by K ( A , B ) the set of all subgroups C of B such that C H A = 0 and A + C/o is S-pure in B / ^ , i . e .

K O ( A , B ) = ^ C C B , s u c h t h a t ( i ) C ^ A = 0 , and ( i i ) A + C/o i s S - p u r e

S t

in B/pj.

In view of the first condition the second condition just means that if p ^ + c = a €. A where p £ S, b é^ B and c é C implies that

p'^a' = a for some a' <£ A. If D C C g Kg(A,B) then D e Kg(A,B) . Also, since 0 £ K O ( A , B ) just means that A is S-pure in B, we observe that the set K O ( A , B ) is not empty if and only if A is S-pure in B. VJe also note that the subgroups in K O ( A , B ) are just the kernels of those homomorphisms f defined on B for which f(A) is S-pure in f(B) and which induces a monomorphism on A, for, if Ker(f) = C then by the definition, A Ti Ker(f) = 0 and A + Ker(f)/.r^ / „\ is S-pure in ^A<:er(f)' ^^^°® ^^Ker(f) ^ ^*^^-*' ^^^^ ^^ S-pure in f(B) and f induces a monomorphism on A.

Definition 6.1. A group B is an S-pure essential extension of A if A is a subgroup of B and 0 is the only subgroup of B in K O ( A , B ) ,

i.e. if A is S-pure in B and if f is any homomorphism defined on B which induces a monomorphism on A such that f(A) is S-pure in f(B), then f is a monomorphism. If S does not contain any element then the S-pure essential extension is just the essential extension of A; if S consists of all the primes then S-pure essential extension is pure essential extension.

Lemma 6.2 j^S"] , If A i^ S-pure subgroup of B and B i^ S-pure subgroup of C then if C is^ S-pure essential extension of ^^ then B i_s S-pure essential extension of A and C is S-pure essential extension of B.

Lemma 6.3 [ 9] . If B is_ S-pure essential extension of A and A is S-pure in C where C is^ S-pure injective then the identity homomorphism

31,

of A is extendable to a monomorphism from B into C.

Levimia 6.4 Q)"] . If A is S-nnre subgroup of B then there exists a homomorphism f defined on B which induces a monomorphism on A and which is such that f(B) is_ S-pure essential extension of f(A) .

Definition 6.5. A group B v/ill be called maximal S-pure essential extension of a group A if B is S-pure essential extension of A and if B C C and G is S-pure essential extension of A then B = C.

Definition 6.6. A group B will be called S-pure maximal if it is maximal S-pure essential extension of itself, i.e. if C is S-pure

essential extension of B then B = C,

Theorem 6.7. Any group A possesses a maximal S-pure essential extension.

Proof. Suppose A does not contain a maximaal S-pure essential extension. Then, for any ordinal k, one can construct a well ordered sequence A = A „ , A, , A„ , A where i -C j implies that A . is properly contained in A. v;here for each i, A. is S-pure essential extension of A. By Theorems 5.10 and 5-11, there exists an S-pure injective group B such that A is S-pure in B. Let us choose k so that the cardinality of A corresponding to k is greater than the

cardinality of B. But this leads to a contrad.iction, since by Lemma 6.3, there is a monomorphism from A, into B. Hence A possesses a ma-ximal S-pure essential extension.

Theorem 6.8. If B is a maximal S-pure essential extension of a group A and if B C C where A is S-pure in C then B is a direct summand of G.

Proof. If A is S-pure in C, by Lerima 6.4, there exists a homo:n.orphism f defined on C which induces a monomorphism on A and which is such that f(c) is S-pure essential extension of f(A). Since A is S-pure subgroup of C and f is a mononorphism on A, f(A) \,rill be S-pure subgroup of f(C); also since E C C , f(B) C f(C), f(A) will be S-pure subgroup of f(B). Aga-in, since f is a monomorphism on A, A o(Ker(f) H B ) = 0 and since, f ( A ) ^ A + (Ker(f) n, B)/^.^^^^^ ^ ^, and f(B) 3^ ^/j^eUf) r^ B' ^^®^( ^) ^'^- ^ ^ Kg(A,B) . Since B is S-pure essential extension of A, Ker( f) «^ B = 0. Therefore, f induces a monomorphism on B. Since B and therefore f(B) is a maximal S-pure essential extension of A, f(B) = f(c) and therefore C = B @ Ker(f).