Ring extension theory

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RING EXTENSION THEORY

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Dit proefschrift is goedgekeurd door de promotor:

Prof. Dr F. Loens tra

RING EXTENSION THEORY

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT. KRACH-TENS ARTIKEL 2 VAN HET KONINKLIJK BESLUIT VAN 16 SEPTEMBER 1927, STAATSBLAD No. 310 OP GEZAG VAN DE RECTOR MAGNIFICUS Dr O. BOTTEMA, HOOGLERAAR IN DE AFDELING DER ALGEMENE WETENSCHAPPEN, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 17 APRIL 1957, DES NAMIDDAGS TE

2 UUR.

DOOR

LEON AR DUS VAN LEEUWEN

Aan mijn ouders Aan Jo

INTRODUCTION

The purpose of t h i s t h e s i s i s t o d i s c u s s the analogy between the extension theory for groups and the corresponding theory for r i n g s . T h e ' e x t e n s i o n p r o b l e m p o s e d and s o l v e d by 0.Schreier i n 1928 r e a d s ^^ :

Given two abstract groups N and F, find all groups G which con-c

tain N as a normal subgroup, such that /N ^ F. Elements of N will be denoted by 1, n, m, 1 and elements of F by 1, cr, T, p resp. If G is an extension of N by P, we may' choose a set of representatives u^ of the cosets of N in G, corresponding to the elements cr of P under the isomorphism be-tween v N and P, and write G = ^ „ u„N.

Then we have, following 0.Schreier, a factor set (S^,c^ ^) in N with a set of automorphisms S^ of N such that

. 1 s

(1) u u = u c (c e N ) , u n u = n ° ' ( n e N ) ,

cr T err cr,r o-, T cr cr ^ '

(2) c c = c c P . n ^ '^ = c-1 n "^^c

^ ' cr,rp T,p crT,p cr,r' cr, T a,T

and t h e p r o d u c t of two e l e m e n t s u n and u m of G i s given by the e q u a l i t y u n . u m = u c n "^m.

cr r CTT cr, T

So far we have s t a r t e d from a given e x t e n s i o n of N by P and have e s t a b l i s h e d a correspondence between t h i s extension and a s o - c a l l e d f a c t o r s e t ( S ^ , c ^ ^). Conversely, we assume t h a t in the group N a system of e l e m e n t s c^ ^ i s chosen, where CT.T range independ e n t l y over a l l the elements of the group F, anindepend t h a t every e l e -ment cr of F i s a s s o c i a t e d w i t h some automorphism n -* n «^ of N for which c o n d i t i o n (2) i s s a t i s f i e d . Then we can show t h a t t h e r e e x i s t s an e x t e n s i o n G of N by F f o r which t h e g i v e n e l e m e n t s c^ ^ and the given automorphisms correspond to t h i s e x t e n s i o n in the above sense.

In 1934 R.Baer c a r r i e d the t h e o r y somewhat f u r t h e r , using o t h e r methods:

Let us d e n o t e by / I t h e f a c t o r group of t h e subgroup of i n n e r automorphisms i n t h e group o f a l l automorphisms of N. I f now G i s an e x t e n s i o n of N by F and i f t h e element cr of F c o r r e s -ponds t o t h e c o s e t u N of G, t h e n t h e automorphisms of N induced by t r a n s f o r m i n g N by v a r i o u s e l e m e n t s of u^N a r e o b t a i n e d from one a n o t h e r by m u l t i p l i c a t i o n by i n n e r automorphisms; t h a t i s ,

they belong t o t h e same automorphism c l a s s . Thus, t o every e l e -ment cr of P t h e r e c o r r e s p o n d s an e l e m e n t a o f / I and from u^N. u^N = u^,^N and t h e d e f i n i t i o n 'of m u l t i p l i c a t i o n of automor-phisms i t follows t h a t a . a^ = a

cr T CTT

In other words, to the extension G there corresponds a well-defined homomorphic mapping of F into /I which is called a

' K o l l e k t i e f c h a r a k t e r ' of the group F in the group NbyBaer. (cf. Chapter I I I ) ,

Finally, in the case that N is abelian s i g n i f i c a n t progress in the extension theory for groups has been made by the use of the s o - c a l l e d cohomology groups.

In 1942 C.J.Everett provided a ring t h e o r e t i c a l analogue of the 0.Scheier ' E r w e i t e r u n g s t h e o r i e ' for groups in his paper'^. He used the same method for his ring extension theory and his results are quite analogous to those of Schreier for groups.

In 1952 L.Rédei introduced a fundamental new method in his paper*\ There he t r e a t e d the extension theory both for groups and rings with the aid of the s o - c a l l e d 'skew p r o d u c t ' (cf. Chapter I ) . In 1954 L.Rédei investigated the ring t h e o r e t i c a l analogues of the c h a r a c t e r i s t i c subgroups and the holomorph of a group. I t is to be noted t h a t , contrary to a group, a ring has more holo-morphs in general. In t h i s paper Rédei also developed some ana-logues of the extension theory, used by Baer in ^^ (cf. Chapter

I and I I ) .

In Chapter I I I we examine the theory of Baer, applied to rings with the notation of Rédei. In Chapter IV the attention turns to the z e r o - r i n g s , which are the analogues of the abelian groups, examined by Baer. In order to obtain a c l a s s i f i c a t i o n of exten-sions we need the notion of equivalent extenexten-sions.In Chapter V we deduce a ring of a l l non-equivalent extensions of a zero-ring S by an a r b i t r a r y zero-ring R. Here, as in a l l cases, the homo-morphic mapping of R in the group of automorphisms of S (ana-logue to the 'Charakter' of Baer^^) is fixed. The l a s t Chapter contains an a p p l i c a t i o n of the theory t o a zero-ring S with a p a r t i c u l a r choice of the additive group of S.

C H A P T E R I

c. J. EVERETT^' has considered the r i n g t h e o r e t i c a l analogue o f 0. SCHREI-ER'S^) extension theory, i . e . the following problem:

Let R and S be given r i n g s ; the problem i s to c o n s t r u c t a l l r i n g s T, containing an ideal S' isomorphic to S , such t h a t

holds.

This problem i s completely solved by Everett.

We c a l l every s o l u t i o n T of t h i s problem a S c h r e i e r e x t e n s i o n of S by r..

Recently L.RE'DEI^^ has introduced a fundamental method for c o n s t r u c t -ing new s t r u c t u r e s from given ones.The new s t r u c t u r e has been c a l l e d by him the skew product of the given s t r u c t u r e s . He has a l s o de-s c r i b e d S c h r e i e r ' de-s extende-sion theory for r i n g de-s by making ude-se of the n o t i o n of skew product by which one may get a survey over a l l ex-tensions'*^ We s h a l l use Rédei's treatment by the aid of which Ever-e t t ' s Ever-extEver-ension thEver-eorEver-em for r i n g s may bEver-e formulatEver-ed as follows: Let R = 0, a, b, . . . and S = 0, a, /3, . . . be two a r b i t r a r y r i n g s . We consider a Rédeian skew product T = Ro S of the r i n g s R and S. The elements of T are a l l p a i r s (a, a) with a e R, a e S. F u r t h e r addition and m u l t i p l i c a t i o n in T are defined as follows:

(2) (a, a ) + (b, /3 ) = (a + b, [a, b] + a + /3). (3) (a, a ) (b, /3) = (ab, {a, b} + ab + a/3 + a/3 ),

where

(4) [a, b ] , {a, b}, ab, a/3 (e S)

are functions of two v a r i a b l e s with v a l u e s in S and s a t i s f y i n g t h e following conditions:

(5) [0,a] = [a,o] = {a, o} = {0,a} = aO = Oa = Oa = aO = 0.

Such four f u n c t i o n s determine uniquely the skew product T = R o S . Conversely, by (2), (3) and (5) we obtain

(a,0) + (b,0) = (a + b, [a, b ] ) , (a,0) (b,0) = (ab, {a,b}), (0,a) (b,0) = (0,ab), (a.O) (0,/S) = (0,a/S),

therefore the four functions (4) are uniquely determined by the skew product T = R o S . The skew productT= Ro S i s a Schreier extension of S by R obeying (2), (3) and (5) if, and only if, the following conditions a r e s a t i s f i e d for a l l alementsa, b, c e R and a, fi, y e S: (6) a ( ^ + 7) = a ^ + ay. (a + /3)c = a c + /3c ,

(7) (a + b ) y + [a, b] 7 = a y + b y , a ( b + c) + a [ b , c ] = ab + ac, (8) a ^ y = ( a / ; ) y , a/3c = a(/3c),

(9) a b y + {a, b}y= a(by) , abc + a{b, c} = (ab)c, ;10) (a/3) c = a(/Sc),

( a b ) y = a ( b y ) ,

{ab, c} + {a, b }c = {a, bc } + a{b, c }, [a,b] = [ b , a ] ,

[a, b] + [a + b,c] = [a, b + c ] + [b, c ] ,

[a,b]c + {a + b,c} = [ac, bc] + {a, c} + {b, c} , a[b,c] + {a, b + c} = [ab, ac] + {a,b} + {a, c } . These r i n g s T exhaust a l l Schreier extensions of S by R. The elements

(O,a) form an ideal S' of T, which i s isomorphic t o S under the i s o -morphism (0,a) — a; further

T/S' .-^ R ( ( a , 0) + S' - a) holds.

If the functions (4) s a t i s f y c o n d i t i o n s (6) - (15), then the f i r s t two of them are c a l l e d a d d i t i v e and m u l t i p l i c a t i v e f a c t o r system, r e s p e c t i v e l y , while the two l a s t ones are said to be r i g h t and l e f t operator s e t , r e s p e c t i v e l y . I f [a,b] = 0 and {a,b}= 0 for a l l a , b ( e R ) , the extensions equivalent t o T are s a i d t o be s p l i t t i n g extensions of S by R.

In a recent paper^^ L.RÉÏDEI has introduced t h e r i n g t h e o r e t i c a l ana-loguesof the c h a r a c t e r i s t i c subgroupsand of the holomorph of a group. For groups the d e f i n i t i o n s are:

A subgroipof a group P i s a c h a r a c t e r i s t i c subgroup if i t i s a norm-al subgroup in every Schreier extension of the given group P.

The holomorph of a group P i s the s p l i t t i n g e x t e n s i o n of P by the full automorphism groip of P.

With these d e f i n i t i o n s one can prove the following theorems:

A subgroup of a group P i s a c h a r a c t e r i s t i c subgroup if, and only if, i t i s mapped into i t s e l f by a l l automorphisms of P.

A subgroup of a group P i s a c h a r a c t e r i s t i c subgroup if, and only if, i t i s a normal subgroup in the holomorph of P.

The ring t h e o r e t i c a l analogues of these d e f i n i t i o n s are:

A subring of a r i n g P i s c a l l e d a c h a r a c t e r i s t i c subring if i t i s an ideal in every Schreier extension of the given r i n g P.

The holomorphs of a r i n g P are the s p l i t t i n g extensions of P by the maximal rings of r e l a t e d double homothetisms of P. (cf. chapter I I ) . Now the c h a r a c t e r i s t i c subrings are c h a r a c t e r i z e d by the following theorems:

A subring of a ring P i s a c h a r a c t e r i s t i c subring if, and only if, i t is mapped into i t s e l f by a l l double homothetisms of P.

A subring of a ring P i s a c h a r a c t e r i s t i c subring if, and only if, i t is an ideal in a l l holomorphs of P.

The usefuUness of these concepts i s obvious.

The d e f i n i t i o n s and theorems for groups are q u i t e analogous to those of r i n g s if one l e t correspond a maximal r i n g of r e l a t e d double homothetisms of P t o the f u l l automorphism group of P and a double hemothetism of P to an automorphism of P.

By making use of t h e s e analogous n o t i o n s , Rédei o b t a i n s t h a t h i s holomorph t h e o r y for r i n g s i s c o m p l e t e l y analogous to t h e known holomorph theory for groups.

The Schreier extension theory for groups has also been i n v e s t i g a t e d by R.BAER^\ using the c l a s s e s of automorphisms,induced in the given group P by h i s e x t e n s i o n g r o i p . With the aid of t h e double homo-thetisms, defined by Rédei for r i n g s , one can s i m i l a r l y i n v e s t i g a t e the Schreier extension theory for r i n g s .

I t i s our purpose to give an extension theory for r i n g s , analogous t o t h a t of Baer for groups.

CHAPTER I I

Let P be an a r b i t r a r y r i n g . For an a r b i t r a r y double mapping a of P i n t o i t s e l f , t h e images of an element a e P a r e denoted by aa, aa r e s p e c t i v e l y . Thus a e x i s t s of the two mappings a -• aa and a — a a (in t h i s o r d e r ) .

In the set of these double mappings we define the sun a + b and the product ab as t h a t element of t h e s e t , for which resp. (1) (a + b)a = aa + ba, a ( a + b) = aa + ab

(2) a b a = a ( b a ) , a a b = (aa)b holds.

Definition: A double homothetismof P i s a double mapping a of P i n t o

i t s e l f with the p r o p e r t i e s :

(3) a(a + /3) = a a + a/3, (a + /3)a = a a + /3a, (4) aa/3 = ( a a ) / 3 , a/3a = a(/3a), (5) (aa)/3 = a ( a / S ) ,

(6) (aa)a = a(aa)

for all elements a, /3 e P.

Two double homothetisms a and b of P are related, if (7) (aa)b = a(ab), (ba)a = b(aa)

holds for all a e P.

As (aa)a= a ( a a ) , every double homothetlsm of P i s r e l a t e d to i t s e l f . With a s e t or a r i n g of r e l a t e d double homothetisms we mean a s e t resp. a r i n g of double homothetisms, for which every p a i r of elements i s r e l a t e d . A s i n g l e double homothetlsm forms a s e t of r e l a t e d double homothetisms, as i t i s r e l a t e d to i t s e l f . Therefore t h e e x i s t e n c e of s e t s of r e l a t e d double homothetisms i s asserted.

L e t u s mention b r i e f l y the notions connected with the r e l a t e d double homothetisms of P.

Every s e t of r e l a t e d double homothetisms of P i s c o n t a i n e d in a maximal one (maximal in s e t - t h e o r e t i c a l sense).

The r i n g {M}, generated-by a s e t M of r e l a t e d double homothetisms of P , i s always a r i n g of r e l a t e d double homothetisms of P.

Every maximal s e t of r e l a t e d double homothetisms of P forms a ring; we c a l l these r i n g s the maximal r i n g s of r e l a t e d double homothetisms of P.

So every ring (and even any set) of r e l a t e d double homothetisms of P i s contained in a t l e a s t one maximal r i n g of r e l a t e d double homo-thetisms. We omit the proofs.5)

An a r b i t r a r y (not necessary maximal) r i n g D of r e l a t e d double homo-thetisms i s defined by (1) - (7), or e x p l i c i t l y :

D i s a r i n g of double mappings of P i n t o i t s e l f , for which the following c o n d i t i o n s are f u l f i l l e d for a l l a, b e D and a, /3 e P: (8) a ( a + /3) = aa + a^, (a + ^ ) a = a a + /3a.

TECHNISCHE HOGESCHOOL

PRIVATE P R O M O T I E

Ter verkrijging van de graad van doctor in de technische wetenschap zal de heer

LEONARDUS CORNELIS A N T O N I U S

V A N LEEUWEN

woensdag 17 april 1957 des namiddags van twee tot drie uur in de vergaderzaal van de Senaat, Oude Delft nr 118, tegenover een commissie uit de Senaat der Technische Hogeschool een proefschrift en stellingen verdedigen, beide goedgekeurd door de promotor prof. dr F. Loonstra.

De titel van het proefschrift is: ,,Ring extension theory".

De Rector Magnificus, O. BOTTEMA.

(9) (a + b)a = aa + ba, a ( a + b) = aa + ab, (10) a a / S = (aa)/3, a/3a = a(/3a), (11) aba = a(ba), a a b = (aa) b, (12) ( a a ) / 3 = a(a/3),

(13) ^ ( a a ) b = a ( a b ) .

For every r i n g D, as previously defined, we define in the s e t of a l l p a i r s ( a , a ) (a e D, a e P) t h e two o p e r a t i o n s (sum and p r o d u c t )

(a, a) + (b,/3) = (a + b, a + /3),

(a,a)(b,/3) = (ab, ab + a/3 + a/3).

Thuswe obtain by RÉDEI*), a splitting Schreier extension of P by D, which is denoted, by D o P. In this ring the elements (0,a) form an ideal, which is isomorphic to P under the isomorphism (0,a) — a. So we see the importance of the double homothetisms particularly in connection with the Schreier extension theory.

Every element y e P induces a double homothetismof P by the two mappingsa-* y a and a - a y for all a e P, as the conditions (3) - (6) are clearly satisfied. We call this an inner double homothetlsm of P.

Two inner double homothetisms are always related by (7). Moreover, the set of all inner double homothetisms of P forms a ring of related double homothetisms of P by (8) - (13).

This ring of inner double homothetisms is contained in at least one maximal ring of related double homothetisms of P, as this follows from the properties of rings of related double homothetisms.

Conversely, every maximal ring M of related double homothetisms of P must contain the ring of inner double homothetisms ofP : Dg. Siflipose, D is not a subring of the ring M.

As every double homothetlsm of M is related to every double homo-thetlsm of Dp by (4), the set {M, 0^} is a ring of related double homothetisms of P and M c{M, D } from the assumption.

But this contradicts the maximality of M.

(We do not need the entire D ; {M, y} is sufficient for the proof, with y as an arbitrary inner double homothetlsm of P, not in M ) . The ring D of inner double homothetisms of P forms an ideal in every maximal ring M of related double homothetisms of P.

Let a be an arbitrary element of M and y e D-.

Then the product a y is again a double homothetlsm e D and consists in: a — (ay) a; a -> a(ay).

If y and S belong to D^, then y - S belongs to D as D^ is a ring. This completes the prcxjf.

So we can form the ring of residue classes of double homothetisms or the residue class ring of M modulo D , which is denoted by '^/D-. This residue class ring '^/D^ can be considered as a ring theoretical analogue in the Schreier extension theory for rings to the factor group of the groi5)of all automorphisms over the group of inner auto-morphisms of a given group P.

An analogue t o t h e group of inner automorphisms i s formed by t h e r i n g of inner double homothetisms D^, while the group of a l l auto-morphisms corresponds t o the maximal r i n g s Mof r e l a t e d double homo-thetisms of P.

CHAPTER I I I

Let T = Ro S be an a r b i t r a r y S c h r e i e r e x t e n s i o n of the r i n g S by the r i n g R, as defined in Chapter I.The elements of T are a l l p a i r s ( a , a ) with a e R, a e S. The elements ( 0 , a ) form an ideal s ' , w h i c h i s isomorpuic t o S under (0,a) - a.

Every element ( c , y ) e T induces a double homothetlsm of S ' , de-fined by

(O.p) - (c,y)(i.p), (O.p) - ( 0 , p ) ( c , y ) for (O.p) e s ' . We denote t h i s double homothetlsm by A,^ , .

If the double homothetisms A,„ „ , and A,„ , are induced b y e l e -( a , a J J {a,, an)

ments (a,a ) and (a,a ) resp.,which belong to tne same residue class (a,0) + S' of T/s'. then A. ^^^ - A ^ ^ ^ ^ ^ ^ = A(^_ ..^^^ .isan in-ner double homothetlsm of S , as (a,a^) - (a,a2) belongs to S'.Con-versely, if the difference of two double homothetisms of s', induced by elements of T, is an inner double homothetlsm of S', then the elements of T belong to the same residue class of V s ' .

From the associative law in T we obtain, that all double homothetisms of S', induced by elements of T, are related. As the conditions for a ring are satisfied, these double homothetisms of s' form a ring D of related double homothetisms of s'.

As we saw earlier, this ring D is contained in at least one maximal ring Mj of related double homothetisms of S'.

Further, the ring of all inner double homothetisms of S', induced by the elements (0,y) e S', forms a subring D^ in the ring D. Accor-ding to the results in Chapter II, the ring D^ is an ideal in every maximal ring M. Therefore, D. is also ideal in that ring M,, which contains D.

Now every residue class (c,0) + s' of V S ' induces a complete resi-due class of ^ / D Q . Thus with each element a e R, which corresponds with (a, 0) + S', there is associated a definite element A(a) of '"1/0^; i.e. a residue class of a maximal ring M of related double homothetisms of s' modulo the ring D„ of inner double homothetisms of S . This mapping of R onto i/D^ is a homomorphism.

For suppose, that a - A(a) and b - A ( b ) . A(a) = A,^ ^,+ D and A(b) = A^jj_^^+ Dp, where (a,a) and (b,/3) are arbitrary' elements of

the residue classes (a,0) + s' resp. (b,0) + S' of V s ' . Then: A(a) + A(b) = A^^^^, + A^j, + D^

A(a) A(b) = A(^_„) :A(b_^^ +Do_

B"t A(^^„)+ A(^,^^) = A(^ ^ b, [a,b] + a + ^) f ^(^ + b) ^"d A^ A .) = A(^b,f{a,b} + ab + a/3 +a^) ^ ^ ^ a b ) . which shows t h a t

A(a) + A(b) = A(a + b) and A(a)A (b) = A(ab). We c a l l T a Aj, (a) - R - e x t e n s i o n of s ' .

Definition l: The function A(a), which associates to the elements of

R (a e R) uniquely and homomorphic residue classes of double homo-thetisms of S' , is called a C-Character of the ring R in the ringS'. As the analogue of the centre of a group we have for rings the anni-h i l a t o r of a r i n g .

D e f i n i t i o n : All the elements (0,y) e S ' , which have the p r o p e r t y

( 0 , y ) S ' = S ' ( 0 , y ) = (0,0) (zero element of the r i n g S ' ) , form an ideal s ' * in S'; S ' • i s c a l l e d the a n n i h i l a t o r of s ' .

Two elements ( a , a j ) and (a,a2) e T, which belong to the same residue c l a s s (a, Q) + S' of ^ ^ S ' , induce the same double homothetlsm of S' i f and only i f ( a , a ^ ) - ( a , a ) belongs to the a n n i h i l a t o r s'* of S ' .

Proof: If ( a , a j ) - ( a , a 2 ) e S *, then we have from the d e f i n i t i o n :

{ ( a , a ^ ) - ( a , a 2 ) } s ' = S ' { ( a , a j ) - ( a . a 2 ) } = (0,0) or ( a , a j ) S ' = ( a , a 2 ) S ' and S ' ( a . a j ) = S'(a.a^).

Therefore for each element (0,p) e s' holds: the two double mappings (O.p) - ( a , a ^ ^ ( 0 , p ) , (O.p) - ( 0 , ^ ) ( a , a ^ ) and

(O.p) - ( a , a 2 ) ( 0 , p ) , (O.p) - (0,/3)(a,a2)

are the same.The double homothetisms of S ' , induced by the elements ( a , a p a n d ( a , a 2 ) e T , a r e A and A ^ ^ ^ . S o we have: A^^^^^^ = A^^ ^ Conversely, i f A^^^^^^ = A^^^^^^, then ( a , a . i ) ( 0 , p ) = (a,a2)(0,yS) and (O.p)(a.a^) = (O.p)(a.a^) for a l l (O.p) e S ' . This means: ( a , a j ) S' = (a,a2) S' and S ' ( a , a ^ ) = S ' ( a , a 2 ) or

{ ( a , a p - ( a , a 2 ) } s ' = S ' { ( a , a j ) - ( a , a 2 ) } = ( 0 , 0 ) . which shows, that (a, a j ) - ( a , ag) e S'*. This completes the proof.

We can also s t a t e our r e s u l t as follows:

A necessary and s u f f i c i e n t c o n d i t i o n , t h a t two elements ( a , a j ) and ( a , a 2 ) , which belong to the residue c l a s s (a,0) + S' of '^/S', induce the same double homothetlsm of S' i s , t h a t ( a , a , ) and (a,a„) belong

T / /

t o the same residue c l a s s of ^ S *.

Thus there i s a s s o c i a t e d with e a c h r e s i d u e c l a s s A of V s ' * uniquely a double homothetlsm A^ of s ' . This mapping i s again a homomorph-ic one.

For suppose t h a t -4 - A^ and S - A^ for A, B e ' ^ / s ' * . '

\ " A(a,a) + s ' * ^"^ Afi = A(b,^) + s ' * ' *here ( a , a ) and ( b , ^ ) a r e a r b i t r a r y elements of T, not in S ' * .

Then: A^^^^^ + s ' » + ^ ( b , ^ ) + s ' * ~ •*(a,a) + (b,;3) + s ' * A(a,a) + s ' * '^(b,/3) + S'* ~ ^(a, a) (b, ^) + s ' •

As ( a , a ) £ A and (b,/3) e B we have ( a , a ) + (b,/ö) e A + B and (a,a) + (b,/3) e AB

This shows t h a t

A^ + Ag = A^ ^ g and A^ A g ~ ^AB ^ required. 14

Definition 2' The function A^, which associates with the elements

of P (A e P) uniquely and homomorphic double homothetisms ofS'. is called a character of the ring P in S ' .

(Remark: We use the term ' C - c h a r a c t e r ' and ' c h a r a c t e r ' in the defin-i t defin-i o n s ! and 2 as analogues of ' K o l l e k t defin-i v c h a r a k t e r ' and 'Charakter', used by Baer for groups ^^).

For each p a i r of r i n g s S' and P, t h e r e always e x i s t s a t l e a s t one character of P in S', t h a t i s to say the p r i n c i p a l character (Haupt-c h a r a k t e r ) , whi(Haupt-ch a s s o (Haupt-c i a t e s with ea(Haupt-ch element of P the i d e n t i (Haupt-c a l double homothetlsm of S ' . This i s the s p e c i a l case, i f a l l images

A^(0.p).(0.p)A^ = (0,0) (zero-element of s ' ) .

If a S c h r e i e r e x t e n s i o n T of S has been given, as we saw e a r l i e r , then every element of T induces a double homothetlsm of S' (f^ S) and the zero-element (0,0) of T induces the i d e n t i c a l double homo-thetlsm of S ' .

The c o n c e p t s c h a r a c t e r and C - c h a r a c t e r of a r b i t r a r y r i n g s in t h e r i n g . S ' c o i n c i d e , i f and only i f S' c o i n c i d e s with h i s a n n i h i l a t o r S'*, i . e . if S' i s a zero-ring.

Proof: Suppose R i s an a r b i t r a r y r i n g with elements, 0, a, b, . . .

and S' i s a r i n g with elements 0, a, /3. ... and the c h a r a c t e r and C-character of R in S' coincide. According t o the d e f i n i t i o n 1 with each element a e R t h e r e i s uniquely a s s o c i a t e d a residue c l a s s A(a) of double homothetisms of S ' . As the c h a r a c t e r c o i n c i d e s with the C-character, t h i s c l a s s A(a) c o n s i s t s only of one element A^, which i s associated with a e R by the character according to d e f i n i t i o n ' 2 . Therefore: A(a) = A„ + D. = A„, where D„ i s the r i n g of a l l inner double homothetisms of S .

For the inner double homothetisms, induced by elements y e S ' , we have t h e n :

a - y a = g , a - a y = 0 (a e S ' ) .

Consequently, D^ c o n s i s t s only of the i d e n t i c a l double homothetlsm of S ' . For every ring S' we have obviously : D^ '^ ^ / S ' * .

In order t o prove t h i s , we consider the (inner) double homothetlsm of S', induced by a :

p -> ap . p ^ pa .

This depends only on the c l a s s a* = a + S'*, therefore we can asso-c i a t e the double homothetlsm a with the asso-c l a s s a*.The inverse mapp-ing i s also unique. Moreover, the sum and product of two elements of D„ i s mapped onto the sum and the product of the image elements resp.

As, in our ca^e,the r i n g D has e s s e n t i a l l y one element, the residue c l a s s r i n g ^ / S ' * consist's only of the z e r o - c l a s s s ' • or S' coincides with S'*.

Conversely,if S' = S'*, then, as previously shown, D^ c o n s i s t s only of the i d e n t i c a l double homothetlsm of S' (zero-element of D ) .

Therefore we have for an arbitrary residue class A(a) of double homo-thetisms of S', that this class has essentially one element A^ (a e R). The class A(a), associated to a e R, (def.1) is reduced to the double homothetlsm A^ or the concepts C-character and character (def. 2) are precisely the same. This completes the proof.

Now we start again with the ring T as a Schreier extension of S by R. The residue classes of /S * are clearly contained in those of

V s ' , as the ring S'*, as we saw, is an ideal in the ring S'.

Now we can associate with each residue class A of V S ' * the element a e R , that corresponds to the residue class (a,0) + S' of ^/S', if A is contained in (a,0) + S'. Then we obtain a homomorphic mapping of 'T/S'* onto R.

For suppose t h a t A r e s p . B a r e c o n t a i n e d in t h e r e s i d u e c l a s s e s (a,0) + S' resp. (b.O) + S' of '^/S'.

Then A - a and B - b for a, b e R and A + B resp. AB are contained! in the residue c l a s s e s (a,0) + (b,0) + S' resp. (a,0) (b,g) + S'. But (a,g) + (b,Cl) + S' = (a + b,0) + s ' - a + b and

(a, g ) ( b , 0 ) + S' = (ab,g) + s ' - ab, therefore 4 + B — a + b and AB— ab.

If A runs through the r e s i d u e c l a s s e s A^ = A. A . A ... of /S *, which are t o g e t h e r c o n t a i n e d in t h e r e s i d u e c l a s s (a, 0) + S' of

^ / S ' , t h a t i s , a l l the A^ and only these r e s i d u e c l a s s e s are mapped onto the element a e R , then the double homothetlsm A^, a s s o c i a t e d with A according d e f i n i t i o n 2, runs through a l l and only the double homothetisms of A ( a ) , a s s o c i a t e d with a e R . (def. 1 ) . As we saw, A(a) is the residue c l a s s of double homothetisms, induced by the elements of (a,0) + S ' .

Proof: A fixed r e s i d u e c l a s s A^ of ^ / S ' * i n d u c e s e s s e n t i a l l y one

double homothetlsm A^. of S'.

As -4^ e (a, 0) + S', the induced double homothetlsm w i l l be contained i n A ( a ) . Therefore A^ runs only through the double homothetisms of A(a). Choose now an a r b i t r a r y double homothetlsm k,^ x of A ( a ) . This A,., \ i s induced by the element ( a , a ) e (a,0) + S . But ( a , a ) belongs to a r e s i d u e c l a s s , say A^, of / S *, a s the A^ t o g e t h e r

f i l l up the residue c l a s s (a, 0) + S ' .

The residue c l a s s A^ induces only one double homothetlsm A^ of S ' . Consequently : A,„ „v = A. . I t follows t h a t A, runs through a l l the double homothetisms of A(a).

F i n a l l y A^ runs through each of t h e double homothetisms of A(a) exactly once.

Por otherwise A^^^^^^ = A^ and A^^ ^^^ = A^ and 4^ ^ A^. Then we would have: ( a , a ) ' e A^ and ( a , a ) e A^^, and triH two d i f f e r e n t residue c l a s s e s A^ and A^ of V s ' * would have a common element (a,a)(?^(0,g)) and so we have a c o n t r a d i c t i o n .

Now we say, t h a t A^ i s an a b s o r p t i o n in A(a) a c c o r d i n g t o t h e following

Definition 3: Let A(a) be a C-character of R in S ' , A. a character

of P in S'(A e P), then A. is called an absorption in A(a), if there exists a homomorphic mappingrj of P onto R with the properties: i**; A, is a double homothetism of 1\\TJ(A)\ and

2™': To each double homothetism A of A(a) there exists one and only one A with 'r](A) = a and A^ = A.

Not any character can be o b t a i n e d by a b s o r p t i o n in a s u i t a b l e C-c h a r a C-c t e r .

Theoreml: The character A. of V in S' can be obtained by absorption

in a C-character if, and only if, there exists an ideal I in P such that

1. A„ for C € I is an inner double homothetism of S and

2. there exists to each inner double homothetism A of S exactly one A in I with A. = A.

Proof: Suppose t h a t A^ can be o b t a i n e d by a b s o r p t i o n in a

C-char-a c t e r , nC-char-amely the function A(C-char-a) of the r i n g R(C-char-a e R) in the r i n g S ' . Let 7] be t h e homomorphic mapping of P onto R according t o d e f i n i t i o n 3. The elements of P, which are mapped by rj i n t o t h e z e r o - e l e m e n t of R, form an ideal I in P.

It follows from the d e f i n i t i o n of C-character t h a t the zero-element 0 of R i s associated with the zero-elemijnt A(0) of the residue c l a s s r i n g of double homothetisms of S ' , since t h i s mapping i s a homo-morphic one. But the zero-element of the r e s i d u e c l a s s r i n g i s t h e ideal of a l l inner double homothetisms of S'..Thus we have:

1 i s the s e t of a l l elements A of P with r](A) = 0; A(0) i s t h e s e t of a l l inner double homothetisms of S ' .

According to d e f i n i t i o n 3 we s t a t e

1. A(^ for C e I i s a double homothetism of ^[r](C)] = A(0) = s e t of inner double homothetisms of S' or

A^ for C e I i s an inner double homothetism of S ' .

2. To each inner double homothetism A of A(0) t h e r e e x i s t s one and only one A with 7](A) = 0 i . e . A in I and A^.= A.

Consequently the conditions 1 and 2 are necessary.

Conversely, l e t the conditions 1 and 2 of t h e theorem be s a t i s f i e d . By the character of P in S' there i s a s s o c i a t e d with each element A e P a double homothetism A^ of S' and I i s an i d e a l in P. If the double homothetisms A, and Ap are induced by elements A and 6, which belong t o the same residue c l a s s R of / I , then A^ - A„ = A^.^ i s an inner double homothetismof s ' , as >1 - B e I and A^ for C e I i s an inner double homothetism of s ' according to c o n d i t i o n l.The con-verse i s a l s o t r u e .

If the difference of two double homothetisms A, and A, i s an inner 17

double homothetism A, then there e x i s t s according t o c o n d i t i o n 2 one element A e I such t h a t A. - A, = A = A4 or A. , = A. = A. There i s exactly one A e I with A^ = A, t h e r e f o r e Aj - Aj^ = A. The double homothetisms A. and A, are induced by elements of P, which belong to the same residue c l a s s R modulo I.

The residue c l a s s R of V I induces consequently a complete r e s i d u e c l a s s o f double homothetisms A(R). Moreover we have, if R and R' are two residue c l a s s e s of ^/l.

A(R + R ' ) = A(R) + A(R') and A(RR') = A(R) A ( R ' ) .

For: A. e R - A. e A(R) and A„ e R' - A. e A ( R ' ) .

1 ri . Z A 2

Then R = Aj + I and R' = Aj + I and A(R) = A^ + Do, A(R') = A^ + Do, where D^ i s the ring of a l l inner double homothetisms of S .

(^A^ +Dp) + (A^^ +Dp) = A^^^,^ . D p = A ( R . R ' ) ( A , / Dp) ( A . ^ . D p ) = A , ^ , ^ . D p = A ( R R ' ) . Consequently A i s a C-character of V I in S'.

Now we have to show t h a t the character A, of P in S' can be obtained

A

by absorption in this C-character.

If A e P runs t h r o u g h a l l the elements of the residue c l a s s R of / I , then A. runs through a l l the double homothetisms of A(R) and w e l l exactly once.For suppose A e R , t h e n A^ e A(R). Choose an a r b i t r a r y double homothetism A of A(R). As A(R) i s a residue c l a s s of double homothetisms, A - A^ i s an inner double h o m o t h e t i s m . According to condition 2 of the theorem there i s e x a c t l y one A'e I , such t h a t A - A^ ~ A^' or A = A ^ ^ ^ ' . This shows t h a t A^ runs through a l l the double homothetisms of A(R). There i s also only one element A + A', for A = A^^^/ and A = A^^^" would give A' = A" (e I ) . This s t a t e s , t h a t A^ runs through the double homothetisms exactly once.

In order t o use d e f i n i t i o n 3, we define a homomorphic mapping T] of P onto ^ / I by a s s o c i a t i n g with the element A e P the residue c l a s s A + I of P / I .

Then the character A^ of P can be obtained by a b s o r p t i o n i n A(R) defined above, for

lSt_ A^ i s a double homothetism of A[77(A)] = A(A + I ) = A^ + Dp, as we saw e a r l i e r .

2nd_ To each double homothetism A of A(R) t h e r e e x i s t s one and only one A with TJ(A) = R and A^ = A.

This follows from the fact, t h a t 17 (A) = A + I = R.

So we have to prove: t o each double homothetism A of A(A + I) t h e r e e x i s t s one and only one A with A^ = A. But t h i s h a s been p r o v e d , since A^ runs exactly once through a l l the d o u b l e h o m o t h e t i s m s of A(R) = A(A + I ) . This completes the proof of Theorem 1.

Definition 4: Is for i = 1,2 the character AJ^^o/P^ in S' an

ab-sorption in the C-character A(a) of the ring R in the r i n g S ' by the homomorphic mapping -q^ of P^ onto R, then the two absorptions are called isomorph, if there exists an isomorphic mapping y of P, onto P2 with

1. 7)2 [yfAj] = T^jfA) for all A e P^ and

2. A ( 2 ) = fid)

\(A) ^A •

Theorem 2: Each C-character can be absorbed in one and, within

iso-morphisms, only one manner.

Proof: I s A(a) a C-character of the r i n g R in the ring S', then l e t

Q be the s e t of the p a i r s [a,A^] with a e R and A^ e A(a).

We s t a t e , t h a t [n.A^ = [b,Ajj] i f and only i f a = b and A^ = A^ F u r t h e r i s :

[ a , A j + [b,A^] = [a + b, A^ + Aj,] and [ a . A j [ b , A ^ ] = [ab,A^A^]. F i n a l l y qr^ ^ i = A^. Then we have:

Theorem 2a: Q is a r i n g , q[^^ ^ ] is a character of Q in S ' , which

is an absorption in the C-char&ter A(a); o homomorphic mapping of Q onto R is given 6 y p ( [ a , A ] ) = a.

Proof: The sum and the p r o d u c t of two elements of Q, as defined

above, i s again an element of Q and uniquely defined.

For: A^ + Ajj e A(a) + A(b) = A(a + b), so [a+ b,A + A^ i s anelement qfQ.Is [a,A J = [ a ' , A ^ ' ] and [b,A^] = [b'.Aj,/], then [a, A J + [b, A^,] = [a',A 1] + Lb',A^/]. For: a = a' and b = b ' , so a + b = a' + b' and _{a b} A = A /, A. = A./, SO A + A. = A / + A. I. _{a a ' b b ' a b a b}

Analogous remarks hold for the p r o d u c t .

[a + b, A^ + A ] = [b + a, A + A J , for a + b = b + a ( i n R ) and A + A^ = A. + A , as the addition of double homothetisms is

com-a b b com-a' mutative.

Therefore the addition in Q i s commutative.

As the a s s o c i a t i v e law holds i n R and the a d d i t i o n of double homo-thetisms i s always a s s o c i a t i v e , the a d d i t i o n in Q i s a s s o c i a t i v e . Further: [o,An] + [a,A„] = [a, A„ + A„] = [a,A„], if A. i s the

iden-U a iden-U H a, iden-U tical double homothetism of S .

As we saw earlier, A e A(0) = set of all inner double homothetisms of S'. As a special case, the identical double homothetism is always an inner one.

For each element [a, A J there exists one and only one element [x, A^^], such that [a,A ] + [x, A ] = [o,Ap].This solution x is denoted by -a and A. is the inverse double mapping of A^; i.e. A^ + A ^ " ^0 ° identical double homothetism of S'.

For the multiplication in Q similarly the associative law and the distributive laws hold, as this is also the case in R and for the multiplication of double homothetisms. Consequently: Q is a ring.

F u r t h e r : qr^ ^ ] a s s o c i a t e s with each element [a,A^] u n i q u e l y a double homothetism of s ' : A

If [ a , \ ] = [ a ' , A ^ ' ] , then A^=A^' and therefore q[a, A ] = ^[a',A <]• Moreover t h i s mapping i s a homomorphic one. For: ^ ^ ^ [ a , A j + [b,A^,] = 'lU + b. A^ + A,] = Aa + \ = « I ^ A J * ^ [ b . A j

^ [ a , A j [ b , A , ] = 'ï[ab,A^Aj = V b = ^[a, A J <l[b. A,]. This shows, t h a t qr^ ^ j i s a c h a r a c t e r of Q i n S ' .

Then: /3([a, A J ) = a i s a homomorphic mapping of Q onto R. For: p([a..Aj + [b.Ajj]) = yO([a+ b, A^ + A^]) = a+ b =

p ( [ a . A j ) + /9([b,Aj^])

p( [a, A J [b,Aj,]) = p( [ab, A^A J ) = ab= p( [a, A J )p( [b, A J ) .

With t h e aid of the homomorphic mapping p the C-character A(a) can be absorbed by the character q.

In order t o prove t h i s , we have to show according to d e f i n i t i o n 3: 1: qr , i i s a d o u b l e homothetism of A[/3( [a, A^])]. This i s true,

for q[^^^ ] = A^ and A [ p ( [ a , A j ) ] = A(a).

2: To each double homothetism A of A(a) t h e r e e x i s t s one and only one [a, A J w i t h / 0 ( [ a , A^] ) = a and q[a,. A ] ° A.

Por: In Q occur a l l p a i r s [a,A„] with A„ e A ( a ) . So the given A of A(a) o c c u r s a t l e a s t once i n t h e form La,Aj, as for any element

[c,Ag] in Q holds: Aj.eA(c). This element [a,A] s a t i s f i e s moreover: p([a..A]) = a and qr^ j^i = A. Consequently t h e r e e x i s t s one element

with the r e q u i r e d p r o p e r t i e s . There e x i s t s a l s o only one such an element, por suppose [ a,A ] and [b,Aj,] s a t i s f y both. Then we would have: p([a.Aj) = p ( [ b , A j ] ) = a and q|-^^^ j = q[fj_^ ] = A. or a= b

and A ~ Av^ = A.

But then l a , A ] = [b,Ajj], as we have seen. This completes the proof of Theorem 2a.

Theorem 2b: Each absorption in the C-character A(a) i s isomorphic

with the absorption, given by the character q[^ f^ ] o / Q in S ' . If t h i s has been shown, we have, t h a t every p a i r of absorptions in the e - c h a r a c t e r A(a) i s isomorphic. Consequently, we have proved then Theorem 2.

We s h a l l suppose, t h a t A^ i s a c h a r a c t e r of P in S' and an absorp-t i o n in absorp-the C-characabsorp-ter A(a) wiabsorp-th absorp-the aid of absorp-the homomorphic mapp-ing T7 of P onto R.

Then we s t a t e y(A) - \ri(A). A ] and we wish to show, t h a t y provides an isomorphism of the two absorptions.

As A^ i s a double homothetism of A[7](A)], y has been defined for a l l A e P . We have: \r](A), A ] is for each element A of P an e l e

-ment of Q. I f ^x'^1' ^^^'^ ^ ( ^ i ) = 'n^^'2} ^^^ ^A ' ^A ' ^°^ ^°^*^ 70

are homomorphisms. Therefore y has been uniquely defined for a l l A in P. F u r t h e r , i f y(A^) = y(A^.). then [r](A^).A^] = [T7(A^),A^ ] , consequently 7](A^) = r](A^^) and A^ = A^ . But we have assumed, t h a t A. i s an absorption in A by rj. t h e r e f o r e to each double homothetism A of A(a) t h e r e e x i s t s one and only one element A with 77(A) = a and A = A. In A[77(A)] each double homothetism occurs e x a c t l y once, which implies A. = A. . Therefore, the inverse mapping of y is also

unique.

In the s e t of p a i r s [77(A), A^] with A e P e a c h element of Q occurs. Por: each double homothetism A of A(a) for a r b i t r a r y a e R must occur under the A. with 77(A) = a. So i f we have an a r b i t r a r y e l e -ment [a,A^] of Q, then A^ e A(a) and t h e r e e x i s t s a t l e a s t one element A e P with 77(A) = a and A^ = A^. But t h i s occurs e x a c t l y once, and t h e r e f o r e the element A e P i s uniquely determined and [a,A^] = [77(A),A,]. Now we can s t a t e : y i s a one-to-one mapping of the whole P o n t o the whole Q.

y is also an isomorphy of P onto Q, for

y(A, + A2) = [77(Aj + A2),A,^^^^] = [77(Aj) + 77(A2),A,^ + A^^] = [ ^ ( ^ l ) . A ^ ] + [77(A2),A^ ] = y(A^) + 7 ( A 2 ) ,

7 ( ^ 4 / 2 ^ = h('4j'42>'A^ A ^ = h('4i)77(A2).A^ A_^ ] = [7?(Aj),A^ ][77(A2).A^ ] = y(A^)y(A2),

as 77 i s a homomorphic mapping and A i s a character.

F i n a l l y . y provides an isomorphism of both absorptions, as according t o d e f i n i t i o n 4,

1. piy(A)] = /0([77(A),A^]) = 77(A) for a l l A e P

2- V'») = '>h(^).Aj=

\-The C-character A(a) of R in S' provides also a C-character of R in S ' / S ' * .

The elements of S' are denoted by 0, a, fi those of R byO, a, b , . . . and l e t A be an a r b i t r a r y double homothetism of s ' (A e A ( a ) ) . Por a l l double homothetisms of S' we have t h a t S'* i s mapped i n t o i t s e l f .

A c o n s i s t s of the double mapping: a — Aa; a — aA. If /3 e S'*, then Afi, /3A e S'* for a l l /3 e s ' * .

Por: (A/3) y = A(/3y) = A(0) = g and

y(/3A) = (y/3)A = (g)A = 0 (y a r b i t r a r y in S ' ) . y(A/3 ) = (yA) /3 = g

(/3A) y = /3(Ay) = 0 (y arbitrary in ^ ' ) .

Now, if we have a residue class a + S'• of ^ /S'*,then A(a + S'*) = A(a) + A(S'*) and (a + S'*)A = (a)A + (S'*)A. Therefore the double

/ s ' h o m o t h e t i s m A of S p r o v i d e s a d o u b l e h o m o t h e t i s m o f / S ' * ; f o r e a c h a + S ' * e ^ ' / S ' * : a + S ' * - A(a) + S ' * ; a + S ' * - (a)A + S ' * . T h i s mapping d o e s n o t depend on t h e c h o i c e o f t h e r e p r e s e n t a t i v e a. f o r i f /3 e a + S ' * , t h e n /3 = a + y w i t h y e S ' * and A(/3) = A ( a ) + A ( y ) e A ( a ) + S ' *, So we h a v e : /3 + S ' * - A(/3) + S ' * = A(a) + S ' * and t h e same h o l d s f o r t h e o t h e r image e l e m e n t . Por e v e r y r e s i d u e c l a s s of / S * we can a p p l y now t h e double homo-t h e homo-t i s m A of s ' .

The image e l e m e n t s a r e a g a i n r e s i d u e c l a s s e s of / S • and u n i q u e l y d e t e r m i n e d . T h e r e f 9 r e t h e d o u b l e h o m o t h e t i s m A o f S' p r o v i d e s a double mapping of / S ' * i n t o i t s e l f . T h i s d o u b l e mapping s a t i s f i e s a l s o t h e c o n d i t i o n s (3) - (6) of C h a p t e r I I : A(aj + S ' * + a 2 + S ' * ) = A ( a j + a 2 + S ' * ) = A ( a ^ ) + A(a2) + S ' * = ( A ( a j ) + S ' * ) + (A(a2) + S'*.). ( a j + S'* + a^ + S ' * ) A = (a^ + a 2 + S ' * ) A = ( a j ) A + ( a 2 ) A + S ' * = ( ( a j ) A + S ' * ) + ( ( a 2 ) A + S ' * ) . a s A ( a j + a 2 ) = A ( a ^ ) + A(a2) and ( a + a )A = , ( a )A + ( a )A. A ( ( a j + S ' * ) ( a 2 + S ' * ) ) = A(aja2 + S'*) = A(aja2) + S ' * = ( A ( a j ) ) a 2 + S ' * = (A(aj) + S ' * ) ( a 2 + S ' * ) .

((a^ + S'*)(a^ + S ' * ) ) A = (a^a^ + s ' * ) A = (a^a2)A + s ' * = ' a j ( ( a 2 ) A ) + S ' * = ( a ^ + S ' * ) ( ( a 2 ) A + S ' * ) , a s A ( a ^ a 2 ) = ( A ( a j ) ) a 2 a n d ( a j a 2 ) A = a ^ ( ( a 2 ) A ) .

( ( a j + S ' * ) A ) ( a 2 + S'*) = ((a^)A + S ' * ) ( a 2 + S'*) = ((a^)A)a2 + S' • = a j ( A ( a 2 ) ) + S ' * = ( a j + S ' * ) ( A ( a 2 ) + S ' * ) = (a^ + S ' * ) ( A ( a 2 + S ' * ) ) , a s ( ( a ^ A ) a 2 = a j ( A ( a 2 ) ) .

(A(aj + S ' * ) ) A = ( A ( a j ) + S'*)A = ( A ( a j ) ) A + S'* = A((a^)A) + S ' * = A ( ( a j ) A + S ' » ) = A ( ( a j + S ' * ) A ) , a s ( A ( a j ) ) A = A ( ( a j ) A ) . T h i s c o m p l e t e s t h e p r o o f , t h a t t h e d o u b l e m a p p i n g A of / S * i n t o i t s e I f , d e f i n e d by a j + S ' * - A ( a j ) + S ' * , a^ + S ' • - ( a j ) A + s ' * i s a d o u b l e homothetism o f ^ / S ' * . P a r t i c u l a r l y t h e i n n e r d9uble homothetisms o f ' S ' p r o v i d e a l s o i n n e r double homothetisms of ^ / S ' * . P o r l e t a - y a , a - a y be an a r b i t r a r y i n n e r d o u b l e homothetism y o f ^ S ' . Then we s t a t e , a s a b o v e , f o r the r e s i d u e c l a s s a + S ' * of ^ / S ' • : a + S' * - y a + S' *, a + S' * - a y + S' *. T h i s d o u b l e mapping of ^ / S ' * d o e s n o t depend on t h e p a r t i c u l a r c h o i c e of t h e r e p r e s e n t a t i v e a . F o r i f /3 e a + S ' * i s a n o t h e r r e p r e s e n t a t i v e , t h e n /3 = a + S w i t h S e S ' * a n d y ^ = y a + y § = -ya , /3y = a y + S y = a y . T h e r e f o r e : y/3 + S ' * = y a + s ' * and ^ y + s ' * = a y + S ' * a s r e q u i r e d . 22

The c o n d i t i o n s (3) - (6) of Chapter I I are again s a t i s f i e d foj- the double mapping a + S ' * - y a + s ' *, a + s ' • - a y + S ' • of ® / S ' * i n t o i t s e l f , as t h i s i s t h e case for the double mapping a — y a , a —ay of S' i n t o i t s e l f , since t h i s i s an (inner) double homothe-tism of s ' . This shows, thai; the double mapping of ^ / S ' * i s an inner double homothetism of ® / S ' * , a s (y + S'*)(a + S'*) = y a + S'* and (a + S ' * ) ( y + S'*) = a y + S'*.

Now we r e c a l l t h a t A i s supposed t o be an a r b i t r a r y double homo-thetism of S' in the given C-character A(a) of R in S'.

Therefore: A(a) = A + D , where D i s the s e t of a l l inner double homothetisms of s ' .

Each inner double homothetism of S' provides, as we have seen, an inner double homothetism of / S ' * . So the s e t D„ provides a s e t

q ' . 0

of inner double homothetisms of /S *. But by this method, all

q' .

the inner double homothetisms of / S ' * are produced.Por; each

q ' .

i n n e r d o u b l e homothetism of / S * can be produced by an inner double homothetism of s ' . P o r i n s t a n c e , a + S'* - (y + S'*)(a + S'*), a + S'* — (a ,+ S ' * ) ( y + S'*) i s an a r b i t r a r y inner double homo-t h e homo-t i s m of ® / S ' * . This can be w r i homo-t homo-t e n a s : a + S ' * -^y a + s ' * a + S'* — a y + S ' • and so we see, t h a t t h e inner double tism a — y a , a — a y of S' produces t h e r e q u i r e d double homothe-tism. As we saw ^ e a r l i e r , A provides an uniquely determined double homothetism of ^ / S ' * .

Consequently, t h e r e s i d u e c l a s s A(a) of double homothetisms of S^ p r o v i d e s a complete r e s i d u e c l a s s of d o u b l e h o m o t h e t i s m s o f ^ / S ' * .

Thus we have: with each element a e R there i s associated a residue Q '

c l a s s A(a) of double homothetisms of / S ' * . u n i q u e l y .

This mapping i s a homomorphic one, s i n c e A(a) i s a C - c h a r a c t e r of R in s ' and therefore the a s s c c i a t i o n of the elements of R with the r e s i d u e c l a s s e s of double homothetisms of S' i s homomorphic. Nowwe have obtained,that^ the given C-character of R in s ' provides a C-character of R in ^ / S ' * .

Corollary 1: There exist always A(a)-R-extensions of / S *.

We r e c a l l that^T i s a A(a)-R-extension of ^ / s ' * if T is a Scju-eier extension of ^ / S ' * by R, such t h a t the residue c l a s s of ' / S ' * ,

q ' .

a s s o c i a t e d with the element a e R , i n d i c e s in / S * the r e s i d u e c l a s s A(a) of double homothetisms of / S ' * .

Here the r i n g s R and S' and the C - c h a r a c t e r A(a) of R in S' are given and T with the r e q u i r e d p r o p e r t i e s should be c o n s t r u c t e d . F i r s t , l e t us c o n s i d e r the r i n g Q of theorem 2a. Suppose, I i s the s e t of a l l p a i r s [o, Ap] of Q, then I i s an ideal in Q. Por: [0,An] - [0,AÓ] = [0,Ap - Ap] and [ a , A j [ o , Ap] = [o.A^Ap] e I for a l l La, A^] e Q.

In [o,Ap] Ap i s n e c e s s a r y an inner double homothetism of S ' , for Ap e A(0) = zero-element of the residue c l a s s r i n g of double homo-t h e homo-t i s m s of S' or homo-the ideal Dp of a l l inner double homohomo-thehomo-tisms of S ' . Conversely, i f we have an inner double homothetism A of S ' , then [ 0 , A ] i s an element of I, (A e Dp = A(0)).

Thereforewe have a one-to-one mapping of I onto Dp under [o,Ap] - A . This mapping i s an isomorphism, for [o,A ] + l o , A ' ] = [o,A + A ] and [0,Ap] [0,Ap] = [O.ApAp] .

Thus we can s t a t e I '^ D„.As we saw e a r l i e r , for every r i n g S' we

q ' , ^ ƒ have Dp J^ / S * under: the inner double homothetism of S , induced

by a, — the res:^due c l a s s a + S'* of ^ / s ' * .

Therefore 1% / S ' * under [o,Ap] — a + s ' * where the residue c l a s s a + S ' * induces the inner double homothetism Ap of S ' .

As I i s an i d e a l in Q, an a r b i t r a r y element fa,A^] of Q induc-es in I a double homothetism, d e f i n e d by

[0,Ap] - [a,A^] [0,Ap], [0,Ap] - [0,Ap][a,A ] for a l l [o, Ap] e I . This can also be written: [o,A.] - [ O , A „ A J , L O , A J - [o,A.A„].

In [a,A^J we have t h a t A^ e A ( a ) , t h e given C - c h a r a c t e r of R i n S ' . As we have seen, t h i s C - c h a r a c t e r p r o v i d e s a l s o a C-character of R in ^ / S • and with the same A^ we have:

a + S'* - A,(a) + S'*, a + S'* - (a)A, + S'*

q/ ^ a

i s a double homothetism of / S *.

Now i t follows t h a t , if in I % ^ ^ s ' * the element [0,Ap] of I cor-responds to the element a + S ' * o f ^ / s ' * , the image elements [o,A^Ap] and A^(a) + S'* r e s p . [0,ApAj and ( a ) A ^ + s ' * w i l l a l s o correspond f^r the same A^. For: a l l elements of t h e residue c l a s s a + S'* of ^ / S ' * induce the double homothetism: y — a y , y - y a of S'and s i n c e [o,Ap] - a + s ' * , we have t h a t a y = Apy and y a = yAp. There-fore we have t h a t the residue c l a s s A^(a) + S'* induces the double homothetism: y - (A^(a))y, y - y ( A ^ ( a ) ) or y - A ^ ( a y ) , y - ( 7 A ^ ) a of s ' . This i s the same inner double homothetism of s ' as:

7 \(\7) = (AaAo)7. 7 ^ (7\)^o = " ^ ^ V o )

-So the residue c l a s s A^(a) + s ' * induces the inner double homothetism

A ^ A p O f S ' .

Similarly, we can show that (a)A^ + s'* induces:

7 ^ ((a.)A^)y, y - y ( ( a ) A ^ ) or y - a ( A ^ ( y ) ) , y - (ya) Ag^ or 7 ^ Ap(A^(y)). y - (yAp)A^ or y - ( A p A ^ ) y . y - y ( A ^ A ^ ) .

Consequently if in I ~ S / s ' * : [o,Ap] a + S'*, then [o,A^Ap] -A„(a) + S ' * and [o,A„A„] - (a)A„ + S ' * . Combining t h e s e r e s u l t s ,

a _ U a a

we o b t a i n : La, A^J induces in I the same double homothetism as A^

S » a a i l l y s *•

[0,Ap] - [ a , A j [ 0 , A p ] = [0,A^Ap], [o.Ap] - [ 0 , A p ] [ a , A j = [0,ApA^ ] coincides with

a + S ' * - A ^ ( a + S'*) = A^ (a) + S'*, a + S'* - (a + S'*)Ag = (a)A^ + S'*. Now we a r e able t o construct a A(a)-R-extension of /S'*.

The correspondence [0,A„] — a + S'* d e f i n e s , as we have seen, an

q '

isomorphism between the i d e a l I of Q and the r i n g / S *. Since ^ / S ' * and Q have no e l e m e n t s in common and Q c o n t a i n s an i d e a l isomorphic to / S ' * , the well-known theorem of imbedding''^ l e a d s us t o a r i n g T, which contains ° /S * and which i s isomorphic t o Q such t h a t under t h i s isomorphism we h^ve [0,Ap] — a + S ' * . Then i t follows, as I is an ideal in Q, t h a t / S ' * i s an ideal in T. Two e l e m e n t s [a,A ] and [ b , A ' ] of Q b e l o n g to the same r e s i d u e

Q a u

K,jLaac v^i / I if and only if a = b. Por assume a = b, then [a, A J - [b,A'] = [0,A^ - A^] e I. Conversely if,

[ a . A j - [b,A^] = [a - b, A^ - A^] e I, then a - b = 0 o r a = b . So a r e s i d u e c l a s s of V I i s formed by a l l t h e e l e m e n t s [a,A^] with f i x e d a and t h e r e f o r e A r u n n i n g t h r o u g h A ( a ) . With each element a e R we can a s s o c i a t e t h e r e s i d u e c l a s s [a.A^l + I of V I . This mapping i s o n e - t o - o n e , f o r i f a = b, t h e n [a,A^] + I = [b,A.] + I , as [a, A ] - [ b , A j e I and if [a, A ] + I = [b,A.] + I,

r i r i

t h e n La, A J - Lb, A. J e I and a = b, as we have seen. Moreover we

a 0

have an isomorphism of V I and R as follows: [a, A J + I - a

[b,Aj^] + I - b

[a + b,A^ + A J + I - a + b [ab, A^Aj^] + I - ab.

Consequently, s i n c e Q i s isomorphic t o T and under the same isomer-phism I ^ s isomorphic t o ° / S * as i d e a l s in Q r e s p . T, we have

t h a t '^/^ / S ' * ^ Q/I. But Q/I % R, as was shown, t h e r e f o r e ' ^ / S ' / S ' * Sï R and T i s a S c h r e i e r e x t e n s i o n of / S * by R. According t o Chapter I the elements of T are a l l p a i r s ( a , a + S'*) with a e R , a + S ' * e ^ / S ' * . The e l e m e n t s ( 0 , a + S'*) form an i d e a l of T, which i s isomorphic t o S ' / S ' * under ( 0 , a + S'*) - a + S ' * . For the sake of s i m p l i c i t y we s u b s t i t u t e now for the i d e a l / S * the isomorphic ima^e (0, / S ' * ) .

Then I % (0, ^ / S ' * ) under [0,Ap] - ( 0 , a + S'*).

For the addition and m u l t i p l i c a t i o n in T holds: ( I , ( 2 ) , ( 3 ) ) : ( a . a + S'*) + (b./3 + S'*) = (a + b, [a,b] + a + /3 + S'*) (a,a + S'*)(b,/3 + s'*) = (ab, {a, b} + a b +a/3 + a/S+ S'*).

Suppose now t h a t in the isomorphism Q'^ T we have: [a,A ] - ( a , s ' * ) , where (a,S *) i s a r e p r e s e n t a t i v e of the ^residue c l a s s ( a , S * ) + ' ^ / S * of / / S ' * , a s s o c i a t e d with a e R.As ' / S ' * ^ V I , we must assume t h a t a r e p r e s e n t a t i v e of the residue c l a s s [a,A„] + I, i . e . [a,A„], of ^ I , c o r r e s p o n d s to a r e p r e s e n t a t i v e of the corresponding residue

c l a s s ( a , S ' * ) + ^ / S ' * , i . e . ( a , S ' * ) , For t h e two c l a s s e s we have: [a,A„] + I - a and ( a , s ' * ) + ^ / S ' * - a.

( 0 , a + S ' * ) - ( a , S ' * ) ( 0 , a + S ' * ) , (0,a + S ' * ) - ( 0 , a + S ' * ) ( a . S ' * ) or (O,a + S'*) - ( 0 , a a + S'*), (O,a + S'*) ^ ( 0 , a a + S'*).

By t h e isomorphism Q ^^ T under [a,A„] ( a , S ' * ) and [0,A„] -(O,a + S *) we o b t a i n , t h a t t h e ' i s o m o r p h i c ' double homothetism of I, induced by the element [a,A^] of Q i s :

[0,Ap] - [ a , A j [ 0 , A p ] , [o.Ap] - [o,Ap][a,Aj or [0,Ap] - [ 0 , A / p ] , [0,Ap] - [0,ApAj.

Sowe have, t h a t in the isomorphism Q'^ T, which contains I % °/S'*: [0,A^Ap] - ( 0 , a a + S'*) and [0,ApA^] - ( 0 , a a + s ' * ) .

As we saw e a r l i e r , we have a l s o :

[0,A^Ap] - (0,A^(a) + S'*) and [0,ApAj - (0, (a)A^ + S'*) Consequently:' ( 0 , a a + S'*) = (0,A^(a) + S'*) and

( 0 , a a + S'*) = (0, (a)A^ + S'*) for a l l a e s ' and a e R .

F u r t h e r : the r e s i d u e c l a s s [a,A^] + I i n d u c e s i n I t h e r e s i d u e c l a s s A(a) of double homothetisms of I.For: in the c l a s s [a,A^] + I, a s we have seen, A^ r u n s t h r o u g h a l l and o n l y t h e double homo-t h e homo-t i s m s o f A(a).An a r b i homo-t r a r y elemenhomo-tof [a,A^] + I, say [a,A^ + A ' ] , induces in I:

[o.Ap] - [a,A^ + A'] [0,A|j], [ 0 , A p ] - [0,A ] [a,A^ + Ap'] or [0,Ap] - [0,(A^+X;)Ap], [O.Ap]- [0, Xp(A^ + A ; ) ] .

Here [o,Ap] i s an a r b i t r a r y element of I or Ap i s an a r b i t r a r y inner double homothetism.

This shows t h a t , i f A' runs through a l l t h e inner double homothetism^ A^+ Ap runs through the double homothetisms of A(a) = A^ + A(0). Again under the isomorphism Q,:^T vie may have [o,Alj -^ (0,y + s ' * ) . Then we o b t a i n : t h e r e s i d u e c l a s s ( a . s ' * ) + ^/S' * i n d u c e s i n

^ / S ' * t h e r e s i d u e c l a s s A(a) of double homothetisms of,®/S'*. Let ( a . y + S ' * ) be an a r b i t r a r y e l e m e n t of ( a , S ' * ) + ^ / S ' * . This element induces:

(0,a+ S ' * ) - (a,y+ S ' * ) ( 0 , a + S ' * ) , (0,a+ S ' * ) - (0,a+ s ' * ) ( a , y + S'*) or (0,a+ S ' * ) - ( 0 , a a + y a + S'*), (0,a+ S ' * ) - ( 0 , a a + ay + S' *) or (0,a+ S'*)- (0,A^(a) + Ap(a) + s'*),(0,a+ S'*)- (0, (a)A^+ (a)Ap+ S'*) or (0,a+ S' * ) - (0, (A^^+ Ap) (a+ S'*)), (0,a+ s ' * ) - (0, (a+ S'*)(A^* A'p. ^ Like above, if y + S ' * r u n s through a l l r e s i d u e c l a s s e s of ^ / s ' *

i . e . Ap t h r o u g h a l l t h e i n n e r d o u b l e h o m o t h e t i s m s of S' o r of / S ' * , A„ + A' runs through the double homothetisms of the residue c l a s s A(a) of double homothetisms of / S *.

For, though A_ has been given as a double homothetism of s ' , A provides a l s o , as we have seen, a double homothetism of / S *. Thus t h e s e t {A„ + A'}= A(a) i s the produced C - c h a r a c t e r of R

m ^ / s ' * . „ ,

1 / q ' ,

Nowwe have obtained, t h a t the residue c l a s s of ' ^ /^ *, associated with the element a e R , i . e . the c l a s s ( a , s ' * ) + / S ' * , induces the residue c l a s s A(a) of double homothetisms of ^ / s ' * .

D e f i n i t i o n 5: Two A(a)-R-e.Jc tens ions of S' belong to the same

ex-tension type, if there exists an isomorphic mapping between them, which leaves S' and R fixed elementwise.

Corollary 2< If the annihilator of S' consists only of the

zero-element, then there exists one, and within isomorphy, only one A(a)-R-ejctens ion of S ' .

The exj.stence of such an e x t e n s i o n follows from C o r o l l a r y 1, as here ^ / s ' * = S' i s .

-In o r d e r t o p r o v e t h e u n i q u e n e s s , we assume t h a t Tj and Tg a r e two A(a)-R-extensions of s ' .

Here A(a) i s again a C-character of R in S ' .

As we saw e a r l i e r , e a c h A(a)-R-extension T of s ' induces a c h a r a c t e r of ^ S ' * in S ' , which i s an a b s o r p t i o n in A ( a ) . As s ' * c o n s i s t s of the zero-element only, t h e c h a r a c t e r A^^^^ of "^i/S'* = 1^ in s ' a s well as the c h a r a c t e r A^?^ of ^ 2 / S ' * = Tg in S' can be obtained by

absorption In A ( a ) .

By theorem 2, the C - c h a r a c t e r A(a) of R i n S' can be absorbed i n one and, w i t h i n isomorphy, only one manner.

Therefore, the two absorptions in A(a) are Isomorphic.

Thus,by d e f i n i t i o n 4 , t h e r e e x i s t s an isomorphic mappingy of T^ onto T2, with

1. V^lyiA)] = 7] (A) for a l l A e T.and 2. A(2) = A(i)

'y(A) A '

where -n.(i = 1,2) i s a homomorphic mapping of T. o n t o R and A^^^ resp. A^?A a r e t h e d o u b l e homothetisms of s ' , induced by t h e elements A resp. y (A) of Tj and T .

Let the element A' e S ' . Then the double homothetism AJP of S' i s induced by t h e element A' e S ' . So we have, t h a t A^P i s an inner double homothetism o f S ' . But a c c o r d i n g t o 2, A('2)/ = A J P i s also an i n n e r double homothetism of S ' , induced by t h e element 7 ( A ' ) , Consequently, 7(A') e S' and as the two induced double

homo-thetisms a r e the same, 7 ( A ' ) and A' b e l o n g t o t h e same r e s i d u e c l a s s of ^ / S ' * . As S ' * - z e r o element of S ' , t h i s means, t h a t 7(A') = A'. Therefore, t h e images of the elements of S' by the i s o

-morphism7 of T onto T2,are again the elements of S' and s ' remains elementwise fixed moreover.

Por the remainder of the proof, we have t o show t h a t 7 l e a v e s the elements of R elementwise fixed. This means, t h a t 7 maps a r e s i d u e c l a s s of "^i/S', a s s o c i a t e d with the element a, onto a residue c l a s s of 2 / s ' , which i s associated with a in "2/S'pï R.

Now since T, — T_ by t h e isomorphism y and S' - S' under the same

1 2 I- / T 7 ( T /

Isomorphism, y induces also an isomorphism between I ' S and 2/s under A + S' - y(A) + s ' for A e T^ , This mapping does not depend on the p a r t i c u l a r choice of the r e p r e s e n t a t i v e A, for i f A' e A + s ' , then,A' A e S' and 7(A') y(A) = y(A'A) = A' A e s ' , t h e r e

fore y(A') + S' = y(A) + S ' . Obviously t h i s mapping i s one-to-one and i s an isomorphism.

By theorem 1, we g e t the character A^^^ of T^ i n s ' by a b s o r p t i o n in A(a), if there e x i s t s an ideal I in T,, such t h a t

1. A^]^ i s an Inner double homothetism of S' for A e I and

2. To each inner double homothetism A of S' t h e r e e x i s t s e x a c t l y one A in I with A,= A.

In t h i s c a s e , the ideal s ' of Tj s a t i s f i e s the c o n d i t i o n s 1 and 2 for the i d e a l I:

A^^^ i s an inner double homothetism of s ' for A e S' and f o r a given inner double homothetism A of S' t h e r e i s e x a c t l y one A e S' with A^ = A. If A^' = A^ = A, t h e n A' and A belong to t h e same residue c l a s s of T / s ' * or Tj, i . e . A' = A. as s ' * i s the z e r o e l e -ment.

Therefore, A^^^ can be obtained by absorption i n A ( a ) w h i c h implies' the existence of the ideal I with conditions 1 and 2 and as we have seen in the proof of theorem 1, the i d e a l I is the s e t of a l l e l e -ments of T-, which are mapped o n t o the z e r o - e l e m e n t of R by t h e homomorphism 77^, d e f i n i n g the a b s o r p t i o n in A ( a ) . Thus the i d e a l I is f o r a g i v e n a b s o r p t i o n A ^ j ' of A(a) u n i q u e l y d e t e r m i n e d . Consequently, in our c a s e , S' i s the s e t of a l l elements of T^, which are mapped onto 0 e R by the homomorphism 77 of T onto R. S i m i l a r l y we can show, t h a t s ' i s t h e s e t of a l l e l e m e n t s of Tj, mapped onto 0 e R by 772 of 1^ onto R. I t follows, t h a t 77 resp. •n d e t e r m i n e s a mapping of t h e r e s i d u e c l a s s e s of ^ i / S r e s p .

2 / s ' onto R and t h i s mapping i s an isomorphism.

For an a r b i t r a r y residue c l a s s A+ S' of ' ^ i / S ' we obtain: 77^ (A+ S') = 77 (A) e R, as 77 ( S ' ) = 0. Por t h e c o r r e s p o n d i n g r e s i d u e c l a s s y(A) + S ' o f ' ^ 2 / 8 ' w e have: 772(7(A) + S') = 772(7(A)) e R, as 772(8') = 0. But 77 (A) = 77 (y(A)) f o r a l l A e T , , c o n s e q u e n t l y 77j(A + S ' ) = •^72(7('^) + S ' ) . This means, t h a t the residue c l a s s e s A + S' of i / S ' and y c 4 ) + S' of ' ^ 2 / s ' i n t h e isomorphisms 77^ : i / S ' ; i ; R a n d

772 : "^2/8' % R a r e a s s o c i a t e d w i t h t h e same e l e m e n t of R. C o n v e r s e l y , suppose t h a t f o r t h e r e s i d u e c l a s s e s A^ + S' and A2 + S' of '•'i/S' resp.'^2/s' we have: r]^(A^^ S') =T]^(A^ + S ' ) = a.

Then: r]^(A^) = 772 (A2),but 77^(A^) = 7^2 ( y ( A j ) ) , t h e r e f o r e 772(A2) = -7?2(y(Aj)) or 772(A2 + S') = 'q2(y(A^) + S ' ) , As ^ 2 / 5 ' % R by 772 i s

an isomorphism, we have: A2 + S' = y(A^) + S ' . Consequently: i f the r e s i d u e c l a s s e s A^ + S' of ' ^ l / S ' and A2 + s ' of 2 / S ' a r e a s s o c i a t e d with the same element of R by the isomorphisms TJ^ r e s p . 77-, then A. + S' and A„ + S' are corresponding residue c l a s s e s in

1 2 rp , T / the isomorphism y between ^ i / S and 2/S .

This shows: c o r r e s p o n d i n g r e s i d u e c l a s s e s of " V S ' and ^ 2 / s ' by y - r e s i d u e c l a s s e s of ^ i / S ' and ^ 2 / S ' , a s s o c i a t e d with t h e same

a e R by 77j r e s p . 77^. If we r e p l a c e such c l a s s e s by t h e i r image 28

elements for 77 and 77 i . e . a, we see, t h a t y leaves R elementwise fixed, as required.

This completes the proof.

Corollary 3: If T is a A ( a ) - R - e x t e n s i o n of S'.then '^/S'* is

iso-morphic with Q, such that the residue class of / S ' * , associated with the element [a, A ] of Q, induces the double homothet ism A

in S ' . - Consequently, the extension type of the A ( a ) - R - e x t e n s i o n s is at least 'modulo S'* ' uniquely determined.

Proof: As T i s a A ( a ) - R - e x t e n s i o n of S ' , T i n d u c e s a c h a r a c t e r

of / S ' * i n s ' , which can be o b t a i n e d by a b s o r p t i o n i n A ( a ) . According to theorem 2b, each a b s o r p t i o n in the C - c h a r a c t e r A(a) of R in S' i s isomorphic with the absorption,given by the character q[^ A ] of Q i n S ' . Let A^ be the c h a r a c t e r of V s ' * in S ' , and rj the'hSmomorphism of / S ' * onto R, d e f i n i n g the a b s o r p t i o n A^ in A ( a ) . Then t h e two a b s o r p t i o n s A^ and qPg^ ^ ] ^^^ i s o m o r p h i c . According t o d e f i n i t i o n 4, t h e r e e x i s t s an isomorphic mapping y of Q onto '^/S'* with

1. 77[y[a,Aj] = p ( [ a , A j ) for a l l [ a , A j e Q

2- \ [ a , A j = q [ a , A j .

The r e s i d u e c l a s s of / S ' * , a s s o c i a t e d w i t h t h e element [a,A^] Of Q by y, i s y [ a , A ] and t h i s c l a s s induces the double homothetism *r[a,A ] of S'- B"t A r ^ ^ ] = q [ a , A j = \ ' ^^ we saw e a r l i e r , t h e r e f o r e the r e s i d u e c l a s s y L a , A ^ J o f ^/S • induces the double homothetism A^ of S ' . Por the remainder of the proof we can show t h a t i f T, and T„ a r e two A ( a ) - R - e x t e n s i o n s of S ' , " ^ i / S ' * and ^2/S • belong to the same extension type of / S *. By c o r o l l a r y 1, there" e x i s t always A ( a ) - R - e x t e n s i o n s of ^ / S ' * , and i f T^ i s a A(a)-R-extension of S ' , then ' ^ / S ' * i s a A(a)-R-extension of ^ / S ' * ,

and as we w i l l prove, if Tj r e s p . T2 i s a A ( a ) - R - e x t e n s i o n gf S ' , then ^ i / s ' * and " 2 / 8 ' * belong to the same extension type of ^ / S ' * . This means, the e x t e n s i o n type of A ( a ) - R - e x t e n s i o n s of * / S * i s uniquely determined.

F i r s t , l e t us assume, t h a t T ji.s a A ( a ) - R - e x t e n s i o n of s ' , t h e n T/S'* i s a A(a)-R-extension of ^ / s ' * .

Por, as we have seen, ^ ' s ' * % Q by the isomorphism y. By the same isomorphism t h e r e i s a s s o c i a t e d with t h e element [0,Ap] of t h e i d e a l I of Q t h e r e s i d u e c l a s s y[o,A ] , which induces the inner double homothetism Ap of S ' . But I ft; ^ / S • under [o,Ap] - a + s'*, where t h e r e s i d u e c l a s s a + s ' * i n d u c e s t h e double homothetism A„ of S ' . Consequently, y [ o , A . ] = a + s ' * and I 7^ ^ /S'* by t h e

isomorphism y . Since^ I i s an i d e a l i n Q, "^ / S * i s an i d e a l in ' ^ / S ' * and V s ' * / S / s ' * ^ Q/j f^ R^ where t h e l a s t isomorphism follows from the proof of c o r o l l a r y 1. As we have seen, yla.^A^] +