THE UNIVERSITY OF MI CHI GAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report ON THE CENTRALIZER OF A GALOIS RING Jiang Luh ORA Project 05260 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NO. G-24333 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR May 1963

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1963.

TABLE OF CONTENTS Page ABSTRACT ~ -V-. INTRODUCTION 1 PRELIMINARIES 4 CHAPTER lo SOME LEMMAS 19 2, PROOF OF THE MAIN THEOREM 24 35 PROOF OF THE MAIN THEOREM (CONTINUED) 32 BIBLIOGRAPHY 49 ili

ABSTRACT Let V and W be a pair of dual vector spaces over a division ring Do There is an associated weak topology on V, a subbase at zero consisting of the kernels of the functionals in W. The resulting topological vector space V is said to be weakly topologized. The ring A =-o (VW) of all continuous linear transformations on V is called a continuous transformation ring. When V is one dimensional A is a division ring. When V is a finite dimensional then A becomes a typical simple ring with minimum condition. And when W is the conjugate space of V then A is a completely primitive ring. The classical Galois theory consists in studying the one to one correspondence between the groups of automorphisms of a field and the subfields of invariant elements. Similar theories have been developed for division rings by Cartan and Jacobson, for simple rings with minimum condition by Hochschild and Nakayama, for completely primitive rings by Dieudonne and Nakayama, and for continuous transformation rings by Rosenberg and Zelinsky. In all these cases, except when A is a field, the Galois correspondence does not pair off an arbitrary subgroup with an intermediate subring. One of the main theorems of classical Galois theory states that the intermediate field E is Galois over the base field Eo if and only if the Galois group, F, of A over E is normal in the Galois group, F0, of A over Eo. In proving a generalization of this theorem in the case of continuous transformation rings, Rosenberg and Zelinsky had to make the ad hoc hypothesis that the centralizer of E in A is a semi-simple ring. The principal result established in this thesis is that this hypothesis is not necessary. In fact, one can prove the following theorem Let (V9W) be a pair of dual vector spaces over a division ring D. Let A =- (V,W) be the ring of all continuous transformations on V which is topologized weakly by W, and let E be a subring of A which is also a continuous transformation ring. Denote byY (A) the socle of A (i.e., the sum of the irreducible left ideals of A) and by f'(E) the socle of E Suppose that (i) 6 (E)v = V (ii) WM~(E)* = W

(iv) V is a finitely generated) (E)-module, (v) G is a group of automorphisms of A with [A: DT] < 0o where A is the group of all semi-linear transformations on V belonging to G and T is the group of all linear transformations on V contained in A. Suppose further that (vi) E is the fixed ring,under G. ThentA(E), the centralizer of E in A, is semi-simple. vi

INTRODUCTION Let V and W be a pair of dual vector spaces over a division ring D. There is an associated weak topology on V, a subbase at zero consisting of the kernels of the functionals in W. The resulting topological vector space V is said to be weakly topologized. Let A =O (V,W) be the ring of all continuous linear transformations on V. Such a ring after Jacobson [9] is called a continuous transformation ring. When V is one dimensional, A is a division ring. When V is a finite dimensional, then A becomes a typical simple ring with minimum condition. And when W is the conjugate space of V, then A is a completely primitive ring. The usual Galois theory consists in studying the one to one correspondence between the groups of automorphisms of a field and the subfields of invariant elements (see [1]). Similar theories have been developed for division rings by Cartan [3] and Jacobson [8], for simple rings with minimum condition by Hochschil d [7] and Nakayama [15 1, for completely primitive rings by Dieudonne [5 ] and Nakayama [15 ], and for continuous transformation rings by Rosenberg and Zelinsky [16]. In all these cases, except when A is a field, the Galois correspondence does not pair off an arbitrary subgroup with an intermediate subring. One of the main theorems of classical Galois theory states that the intermediate field E is Galois over the base field Eo if and only 1

2 if the Galois group, r, of A over E is normal in the Galois group, FO, of A over Eo. In proving a generalization of this theorem in the case of continuous transformation rings, Rosenberg and Zelinsky [16] had to make the ad hoc hypothesis that the centralizer of E in A is a semisimple ring. The principal result established here is that this hypothesis is not necessary. In fact, we prove the following theorem: Main theorem: Let V and W be a pair of dual vector spaces over a division ring D, and let E be a subring of A = (v,w) which is also a continuous transformation ring. Denote by b'(A) the socle of A and by E (E) the socle of E. Suppose that (i)'(E)V = V, (ii) WA E)* = W, (iii)'((E)C_(A), (iv) V is a finitely generated AE)-module, (v) G is a group of automorphisms of A with [A: DT] < co, where A is the group of all semi-linear transformations on V belonging to G and T is the group of all linear transformations on V contained in A. Suppose further that (vi) E is the fixed ring under G., TheniA(E), the centralizer of E in A, is semi-simple. In Chapter 1, we will prove some lemmas which will be used in proving the main theorem. The proof of the main theorem will appear in Chapter 2 and Chapter 3.

5 The author is very much indebted for many valuable suggestions to Professor Jack E. McLaughlin, under whose direction this thesis was written.

PRELIMINARIES Let R be a ring. Definition 0.1. A left R-module is a system consisting of an additive abelian group M, and a function defined on the product set RxM having values in M, such that if ax denotes the element in M determined by xeM, acR, then (a+b)x = ax + bx (ab)x = a(bx) a(x+y) = ax + ay hold for any a, b in R and x, y in M. The concept of a right module is defined in a similar fashion. Henceforth, the term "module" without modifier will always mean left module. Definition 0.2. An R-module M is said to be unitary if RM = M. Definition 0.3. A subgroup N of M is said to be an R-submodule of M if RNCN. Definition 0.4. If N is an R-submodule of M, the factor group M/N can be turned into an R-module by defining a(x+N) = ax + N. We call this module the difference module of M relative to N, and it will also be denoted by M/N. 4

Definition 0.5. An R-module M is said to be finitely generated if there exists a finite subset ([x, x2,...,xn) of M such that every element in M can be written in the form maxj + m2x2 +... + m x + alxl + a2x2 +... a x nn nn where the mi are integers, and the ai are in R. Definition 0.6. A module M satisfies the ascending chain condition for submodules if for any increasing sequence of submodules NCN2C. o there exists an integer n such N Nn+ = o n Proposition 0.1. A module M satisfies the ascending chain condition for submodules if and only if every submodule of M is finitely generated (see [10]). Definition 0.1. A module M satisfies the descending chain condition for submodules if for any decreasing sequence of submodules N1DN Do there exists an integer n such that, N = N n n+l Proposition 0,2. -f R is a ring that satisfies the descending chain cond.it+ion for ieft+ (right ) ideals. then any finitely generated unitary B-mo-dule satisfies the des cending chain condi+i.on for submodules (right su.bmodules) ( se [10 j).'filnlition 0.8. Let! arnd M' be R-moduleso A group homomorphism

6 G of M into M' is called an R-homomorphism if for all xEM and all aER, G(ax) = aG(x). If @ is a one to one mapping, it is called an R-isomorphism. If there exists an R-isomorphism of M onto M', then M and M' are called isomorphic modules and we write M'M'. If M = M', an R-homomorphism of M into M itself is called an R-endomorphism. We will denote by EndRM the ring of R-endomorphisms on M, and denote by Hom (M,M') the group of R-homomorphisms of M into M'. Definition 0.9. An R-module M is called irreducible if RM / (0) and -there is no proper R-submodule of M other than (0). Definition 0.10. An R-module M is called completely reducible if it is a sum of irreducible R-submodules of M. Definition 0.11. An R-module M is called a direct sum of the family (M' IEA) of R-submodules of M and we write M = GM\ if any x in M can be written in one and only one way in the form Zx. with x ceM. Proposition 0.3. If M = Z Mh where (Mhl eA) is a family of irreNEA ducible R-submodules of M, then M = M, where (M 15EtA) is a sub5EA family of ([M \chA) (see [12]). If M = MA = Z @ N where M, and N. are irreducible R-submodules of M, then the cardinal number of (My} equals the cardinal number of (Nuj.

7 If both sets are finite, the proof of the result can be found, for examample, in [2]. For the infinite case, the proof is available in [12]. From this fact, we can define the dimension of a completely reducible module as follows: Definition 0.12. If M is expressed as a direct sum of irreducible R-submodules, the cardinal number of direct summands is called the dimension of M over R and is denoted by dimRM. In a particular case, if B is a ring with unit element and D a division subring with the same unit element, then B is a completely reducible D-module whose dimension we write [B: D]. Proposition 0.4. Every R-submodule N of a completely reducible Rmodule M has a complement N'; that is, an R-submodule N' of M exists such that M = N G N' (see [12]). Proposition 0.5. Every R-homomorphic image and every R-submodule of a completely reducible R-module is completely reducible (see [12]). Proposition 0.6. Let M be a completely reducible R-module, E = End M. Then M is completely reducible as E-module (see [12]). R Definition 0.15. Let M be an R-module and let (MI jeA3 be the family of all irreducible R-modules of M, then j MA is called the DEA socle of M and it will be denoted by (M). Definition 0.14. Let M be an R-module. The sum X M5 of all the irreducible R-submodules of M R-isomorphic to a given irreducible R-sub

module N of M is called the homogeneous component of the socle determined by N. It is easy to see that the socle of M and its homogeneous components are fully invariant in the sense that they are mapped into themselves by every endomorphism of M. Definition 0.15. If R is a ring, the socle of R as an R-module is called the socle of R. Note that the socle O(R) of a ring R is the sum of irreducible left ideals of R and so OR) is a left ideal. Every right multiplication x+xa is an endomorphism of R as an R-module. This maps (R) into itself; hence t(R) is also a right ideal. Definition 0.16. An R-module M is said to be faithful if aM / 0 for each a / 0 in R. Definition 0.17. A ring R is called primitive (right primitive) if it admits a faithful irreducible module (right module). Definition 0.18. A ring R is called completely primitive if it is isomorphic to the ring of all linear transformations oan a vector space over a division ring. Clearly, a completely primitive ring is primitive. Proposition 0.7. Let R be a primitive ring and let I, and I2 be non-zero ideals in R. Then I1 I2 7 (0) (see [12]). Definition 0.19. Let R be an arbitrary ring and let )rJbe the set of irreducible R-modules. Then the ideal

9 J(R) = 0 (aER laM = 03 is called the (Jacobson) radical of R. Proposition 0.8. Every element z in J(R) is left quasi-regular; i.e., there exists an element z' in R such that Z + Z' - Z'Z = 0 (see [12]). Definition 0.20. A ring R is called semi-simple if the radical J(R) of R is (0). Definition 0.21. A ring R is called simple if there are no proper ideals in R other than (0}, and R (0. [ Proposition 0.9. (Wedderburn's Theorem). Any simple ring R satisfying the minimum condition for left ideals is isomorphic to the complete ring of linear transformations on a finite dimensional vector space over a division ring (see [2]). Definition 0.22. Let V be a left vector space over a division ring D and let W be a right vector space over D. A mapping f of the product set VXW into D is called bilinear form on V and W if for all v, v1l v2 in V, w, wl, w2 in W and d in D, we have f(Vl+V2,W) = f(v1,w) + f(v2,w), f(v,w1+w2) = f(v,wj) + f(vw2), f(dv,w) = df(v,w),

10 and f(v,wd) = f(v,w)d The bilinear form f is called non-degenerate if f(v,w) = 0 for all veV implies w = 0 and f(v,w) = 0 for all weW implies v = 0. If there exists a non-degenerate bilinear form f on V and W, then (V,W) is called a pair of dual vector spaces relative to f over D. In dealing with a single bilinear form, it is convenient to use the abbreviation (v,w) for f(v,w) and we simply say that (V,W) is a pair of dual vector spaces over D. We shall do this from now on. Definition 0.23. Let V be a left vector space over a division ring D. f is called a linear function on V if f is a linear transformation on V into the left vector space D over D. The set V of all linear functions on V is a right vector space over D relative to the laws of composition: (f+g)v = fv + gv, (fd)v = d(fv) for all veV, f,geV, and dED. The right vector space V is called the conjugate space of V. Let V be a left vector space over D and let V be the conjugate space of V. Then (V,V ) is a pair of dual vector spaces over D with the bilinear form given by.(v,f) = fv, for feV, veV. Conversely, if (V,W) is a pair of dual vector spaces, then there

11 is a natural isomorphism Q of W into V given by M(w)v = (v,w), for all vcV. Therefore, we can consider W as a vector subspace of V Definition 0.24. Given a pair of dual vector spaces V and W,there is an associated topology on V, a subbase at zero consisting of the kernels f of the linear functions f in W. The resulting topological vector space V after Dieudonn~ [4] is said to be weakly topologized by W. W can be then retrieved from the topology as the set of all continuous linear transformations of V into the vector space D over D with D carrying the descrete topology. Proposition 0.10. Let V be weakly topologized by W. If U is an open subspace of V, then U is also closed and has finite codimension. Conversely, if a closed subspace U of V has a finite codimension, then U is also open (see [15]). Let V and W be a pair of dual vector spaces. For any subset S of V and T of W, we use the notation S for (feWI(v,f) = 0 for all veS) and T for (veVj(v,f) = 0 for all feT). Proposition 0.11. Let (V,W) be a pair of dual vector spaces over a division ring D. If U is a vector subspace of V, then the closure 11 of U in the weak topology of V is U Hence, a vector subspace U of V is closed in the weak topology if and only if U = U (see [12]). If (V,W) is a pair of dual vector spaces over a division ring, we will denote by (V,W) the ring of all continuous linear transformations

12 on the weakly topologized vector space V. Proposition 0.12. A linear transformation a on V is continuous if and only if for every f in W the linear function v+(av,f) is again an element of W (see [9]). If this is the case, denote this function by fa so that (av,f) = (v,fa ) and a becomes a linear transformation on W. Definition 0.25. Let V be a left vector space over a division ring D. A ring R of linear transformations on V is said to be a dense subring of the ring of all linear transformations on V if for any natural number k, and any linearly independent vectors vl,...,vk, and any k vectors ul, u2,... uk, there exists an aER such that avi = ui i = 1,2,...,k. Definition 0.26. Let V be a left vector space over division ring D. A linear transformation a on V is said to be of finite rank if the dimension of the image, aV, is finite. Suppose (V,W) is a pair of dual vector spaces. Then we denote by 9(V,W) the set of all continuous linear transformations of finite rank on the weakly topologized space V. Proposition 0.13. The following three conditions on a ring R are equivalent: (1) R is a primitive ring with non-zero socle. (2) R is isomorphic to a dense subring of the ring of linear transformations of a left vector space V over a division ring D con

13 taining non-zero linear transformations of finite rank. (3) There exists a pair of dual vector spaces (V,W) over a division ring D such that R is isomorphic to a subring of o((V,W) containing I (v,w). If R is represented as in (2), its socle is the set of linear transformations of finite rank contained in this ring. If R is represented as in (3), then its socle is'(V,w). Moreover, the socle of R is a simple ring which is contained in every non-zero ideal of R (see [12]). Proposition 0.14. Let V and W be a pair of dual vector spaces over a division ring D. Then any finite subset of t(V,W) can be embedded in a subring of 7(V,W) which is isomorphic to a matrix ring Dn (see [13]). Proposition 0.15. Suppose R is a ring of linear transformations on a left vector space V over D and R is a primitive ring with minimal left ideals (abbreviated to P.M.I. ring). Then V is a homogeneous completely reducible R-module if and only if V = \(R)V (see [16]). Throughout this thesis if B and C are subrings of a ring A, we use cjB(C) denote the centralizer of C in B: the set of all elements in B which commute with every element of C. Proposition 0.16. Let e be the class of all subrings E of A = (V,W) satisfying the conditions (1) E is a continuous transformation ring with socle YE), (2) t(E)V = V, (3) WE) =W,

14 Let 3 be the class of all subrings B of ~ = End(V,+) containing D and satisfying (1*) B is completely primitive with socle TB) (2*) MB)V = V (3*) The left D-dimension of a minimal left ideal of B is finite. Then the correspondence E- 4'(E), B-+ o (B) is a one to one correspondence between ( and 3. If E in C and B in correspond, then (I) Every endomorphism of V commuting with E is continuous. (II) If B consists of all linear transformations on a vector space of dimension ~ then dim V =. E (III) V is an irreducible BE-module (IV) If veV, there is an idempotent e in'(E) with ev = v (V) Every E-submodule of V is closed (see [16]). Proposition 0.17. Under the hypothesis of Proposition 0.16, if B is a simple ring with minimum condition, then [B: D] < X (see [16]). Definition 0.27. A mapping A of a left vector space V1 over D1 into a left vector space V2 over D2 is called a semi-linear transformation if (1) A is a homomorphism of (V1,+) into (V2,+), (2) there exists an isomorphism a of D1 onto D2 such that for all veV1 and deD1, we have A(dv) = o(d)Av. We call a the isomorphism associated with A.

15 Proposition 0.18. Let (V,W) be a pair of dual vector spaces over a division ring D, and let A =o<(V,W). Then every automorphism g of A is of the form a+%-lak with A and A-\ continuous semi-linear transformations on V; and conversely (see [11]). In this case, we say A is a semi-linear transformation belonging to g. Clearly, if X belongs to g, D\ = AD is exactly the set of all semi-linear transformations which belong to g, and, moreover, if G is a group of automorphisms of A, then the set A of all semi-linear transformations on V belonging to some g in G form a multiplicative group. We will call A the group of semi-linear transformations on V belonging to G. Now we define the tensor product of two modules. Definition 0.28. Let V be a right module and W a left module over a ring R. Let F be the free abelian group generated by the pairs (v,w) with veV, weW, and let K be the subgroup of F generated by elements of the form (v,w+w') - (v,w) - (v,w'), (v+v',w) - () - (',w), (vr,w) - (v,rw), (reR). Then the tensor product VRW of V and W is defined as the quotient group F/K, regarded as an abelian group. Definition 0.29. Let R be an arbitrary ring and let D be a commutative ring with identity. Then we shall say that R is an algebra over D if a composition (C, x)+Czx of the product set DXR into R is de

16 fined such that (i) (R,+) is a unitary left D-module relative to the composition (a, x)+cax, (ii) for all oeD and x,yeR a(xy) = (Ox)y = x(ay). Definition 0.30. If V and W are algebras over a communitative ring D with u;nit element, then the tensor product module V.DW is an algebra relative to the multiplication composition viwi vjwj = vivjwiwj i j i,j with vi,vjcV, wi,wjeW. We call this algebra the tensor product of the algebras V and W. Proposition 0.19. Let X be a primitive algebra over a field K having a non-zero socle "(X). Assume that (xl,...,xn) is a finite linearly independent subset of X. Then there exists an element sec(X) such that (sx1,...,sxn) is linearly independent (see [12]). Proposition 0.20. Let B be an algebra over a field K which contains a central simple ideal S and C an algebra over K with a unit element. Assume that, if (b,....bm) is a linearly independent subset of B then there exists an element xeBrBI such that (xbl,...,xbm] is a linearly independent subset of S. Assume, moreover, that (1) be = cb for all beB, ceC, (2) Sc = 0 for ceC implies c = O. Then BC

17 = BR C (see [12]). K Finally, we will define quasi-Frobenius rings and list some of their properties. Let A be a ring with identity which satisfies the minimum condition on left and right ideals. If S is a subset of A, we denote, respectively, by r(S) and I(S) the right and left annihilators of S in A. Definition 0.31. If for each left ideal L and each right ideal R in A.(r(L)) = L, r(~(R)) = R, then A is called a quasi-Frobenius ring. Definition 0.32. If for each proper left ideal L and proper right ideal R in A, r(L) ~ 0, 1(R) 0, then A is called Kasch ring. Clearly, every quasi-Frobenius ring is a Kasch ring. Definition 0.33. Let S be a ring with ring A as two-sided operator domain. A mapping Q of S into A is called an operator - homomorphism of S into A if O(as) = aG(s), o(sa) = G(s)a hold for all aeA, seS.

Definition 0.34. An operator-homomorphism 0 of S into A is called a Frobenius homomorphism if there are no non-zero left ideals and nonzero right ideals of S contained in the kernel of 0. Definition 0.35. If there is a Frobenius homomorphism which maps S onto A with A as two-sided operator ring of S, and, moreover, if A is a Kasch ring, then S is called a Frobenius extension of A. Proposition 0.21. A Frobenius extension of a quasi-Frobenius ring is a quasi-Frobenius ring (see E14 ]). Proposition 0.22. If 3 and are finite dimensional algebras over a field Z and if bothe% and are semi-simple, thenX Bz is a quasi-Frobenius ring (see [6]).

CHAPTER 1 SOME LEMMAS In this section, we prove some lemmas that will be used in the proof of our main theorem. Lemma 1.1. Let R be a ring and let M be a faithful completely reducible R-module. Then R is semi-simple. Proof: If R is not semi-simple, then by Definition 0.20 the radical J = J(R) of R is not (0]. Since M is faithful, JM will be a non-zero R-submodule of M. Write M = SC, aeoc where Sa are irreducible R-submodules of M. Hence, there exists an irreducible R-submodule, say S, of M so that JS / 0, and so there exists an sES such that Js t 0. Since Js is an R-submodule of M and is contained in the irreducible R-submodule S of M, we have Js = S. Thus, s = js for some j in J. Now be Proposition 0.8, a keJ exists such that j + k - kj = O. It would follow that s = s - (j+k-kj)s = (s-js) - k(s-js) = 0, 19

20 a contradiction. Therefore, R is semi-simple. Lemma 1.2. Let (V,W) be a pair of dual vector spaces over a division ring D, let A =ct (V,W), and let E be a subring of A which is also a continuous transformation ring with ((E)V = V, W E) = W, and }'E)C a6'A). Then if V is completely reducible as a DE-module,f (E) is semisimpleo Proof: By Proposition 0.16, End V =CA(E). Then according to Proposition 0.6, V is completely reducible as an e: (E)-module. Hence from Lemma 1.1,&'[E) is semi-simple. Lemma 1.3. Let V be a left vector space over a division ring D and let E be a continuous transformation ring which is a subring of A = EndDV. If'(E)V = V, j6(E)C MA), and, furthermore, V is a finitely generated ~O(E)-module, then AE) is a finite dimensional algebra over the center Z of A. Proof: Let W = V (E) (1) V and W are a pair of dual vector spaces. It is trivial that (v,f) = 0 for all veV implies f = 0. On the other hand, if veV and (v,f) = 0 for all feW, then (v,fs ) = 0 for all feV, seV((E) and so (sv,f) = 0 for all feV, se'E) which implies that sv = 0 for all se$(E) and hence ~E)v = 0. Since (E)V = V, by Proposi

21 tion 0.15, V is a homogeneous completely reducible E-module. Write V =' M. 1 i where the M are irreducible E-modules. Thus, v =.Zmi with miCMi. and hence O = (E)v = E)mi, so tFE.)mi = 0 for all i. Let Ni = (nicMilK E)ni = 03. Then Ni being an E-submodule of Mi either equals ()0 or Mi. If Ni = Mi for some i, it would imply that i (E)Mi = 0 for all i since the Mi are E-isomorphic, and then (E)V = 0, a contradiction. Therefore, we have Ni = 0 for all i. So mi = 0 for all i.;, Hence, v = mi = 0. This 1i~~~~~~~~~ i proves that V and W are a pair of dual vector spaces. (2) Let Ao = (V,W). Then ECAo. If aeE, then there exists a* E* so that (av,f) = (v,fa*) for all veV, fEV*. We will show that fa*EW for any feW. Write f = fis fieV, si E)

22 Then we have fa = (. fisi)a = Z fi(sia ) Since, sia E E)*, fa*cV* {E)* = W. Hence, by Proposition 0.12, aEAo. (3) W E)* = W. Since E is a primitive ring, YE)? is a nonzero ideal of E by Proposition 0.7. By Proposition 0.13, [E))]2D E) and hence [O(E)]2 = YE). Thus, [`(E)*]2 = [~(E)2]* =~(E)* Therefore, W (E) = vE ) YE ) = V*AEK) = W. (4) <(E)c_ AO). If se~(E), then seiA) and hence by Proposition 0.13, dimDsV < mx. Therefore, sE6 (Ao). Now from (1) —(4), we see that V, W, Ao and E satisfy all assumptions of Proposition 0.16. Hence B =fC (E) is completely primitive, where i = End (V,+), so B is the complete ring of linear transformation on a vector space V' over a division ring D'. But by Proposition 0.16, dimDV' = dimEV. Since, dimEV, < o, dimDV'< co, and hence by Proposition 0.9, B is a simple ring with minimum condition. By Proposition 0.17, [B: D] < ca. It is clear that I~A(E)CB. Let U be D-subspace of B generated by o&A(E). Then [U: D]l <,oo say tl,...,tn is a basis for U over D. Thus for any tcAA(E), t = Z diti with dieD and dt = td for all deD.

23 This implies that ddi = did for all dED and all i. Hence each dicZ and (ti) spans 4A(E) over Z, thus [A(E): Z] < o. This completes the proof.

CHAPTER 2 PROOF OF THE MAIN THEOREM The main theorem has been stated in the introduction. Now to prove this, we suppose contrarily that iA(E) is not semi-simple. Then according to Lemma 1.2, (vii) V is not a completely reducible DE-module. Therefore, there is a sequence (Vi) of DE-submodules so that O = VOCV CV2C... where the Vi/Vi1i is the DE-socle of V/Vii We shall show first that this sequence of DE-submodules terminates. By Proposition 0.1, it will be sufficient to show that every E- submodule of V is finitely generated. Since V is a completely reducible E-module, V can be expressed as ( M with each M an irreducible E-module. By (iv) 74' V = Y(E)v + Yt(E)v2 +... + HEE)vm and since for each i ie ri' 7 where Fi is a finite subset of r. Thus 24

25 m V =1 = M M i=l YeFri 7cJri i =1 and hence dim V < o. Therefore, dim EU < oo for any E-submodule U of V; that is, any E-submodule is finitely generated. Hence, we have O = VoC V1C....CVnlCVn = V for some integer n - 2 where Vi/Vi-1 is the DE-socle of V/Vi-.-, Our next step is to establish an h / 0 in Hom (V/Vn-i,V1) so that hkv = Ahv for all cEA, VEV/Vn-1 We remark that since the socle of a module is fully invariant, each Vi is an,,A()A)-submodule of V. Now suppose we have found such an h. Then define hEEndDV as follows: Write V = Vnl U (D-direct) and set h(u) = h(u) for uEU and hVn-l = 0. For A\EA, vn-leVn-l, uEU, we have then Ah(vn-l+u) = khu = hku = h(2u) = h(Au) = hk(vn-l+u),

26 so \h = hA for all \cEA and h / 0. Consider the set X = (aEElaUCVnl and aVnl = 0>. It is easy to see that X is a left ideal of E and X2 = 0. Since E is primitive, by Proposition 0.7, X = 0, so h/E. Hence there exists uEU with hu / 0. Also by Proposition 0.16, there exists an idempotent e in J'E) such that eu = u. Thus he / 0. But heV c hV = O. n-i — n-1 By the same argument as above, we have herE. Since eEY E), dimDeV < oo. Let (vl,v2,...,vt be a D-basis for eV. By the density of'A) in EndDV, there is an s in (A) so that svi = hvi i = 1,2,..,t. t Hence, for any vEV, ev = divi i=l with dieD, and then hey = h diVi = ddhvi i=l i=l t t - disv = s dv. = sev. i=l i=l Thus, he = seej A) and hence heEA. Furthermore, for any AeA,

27 (he)- = h(eX) = h(Xe) = (h\)e = (?h)e = A(he), so heiA(A)A But he/E, whence ~A(A)~E properly, orE is not the fixed ring under G which contradicts our assumption (vi). This proves our main theorem except that we have yet to show the existence of such an h / 0 in HomD(V/Vn-L,V1) with h\v = \hv for all ehA, veV/Vn_.i We will proceed to do this by mathematical induction on n under the assumptions (i)-(v) and (vii). The proof of the case n = 2 will be postponed to Section 3. We suppose now that n > 2. Consider the vector space V/V1 and V1+. The bilinear form on V/V1 and V11 given by (v,f) = (v,f) for vEV/V1, fEVl1 is non-degenerate since (vf) = 0 for all fEVl- implies veV1l- = V1 or v = O in V/V1 and (v,f) = 0 for all veV/V1 implies (v,f) = 0 for all veV so f = O. Hence, (V/VilVlL) is a pair of dual vector spaces over D. Let Al =o~(V/Vi,VIL). Then for any a eE 9 (Vlivjla*) =(aV1,Vl-) C(Vl,V1-) = 0. This means that V9la*CV1- so ECA1. Now we are ready to verify the conditions (i) —(v) and (vii) for V/V1 and Vl11 (if)'(E)V/V1 = V/Vl. Clearly, ['E)V/V1CV/V1. On the other hand, since ~(E)V = V, for any veV, there exists vieV, siEC (E) (i = l1,..,k), so that k v = L sivii-=1 k Thus, v = sivi in V/V1, and hence V/V1CE)V/V1. i -l

28 (ii) VC1 (E)* = V-l. Since W = W(E), for any fEW, r f j= ) fisi with fieW, siEiE). By proposition 0.14, there exists i=l an idempotent element e in YE) such that sie = si and si = si e for all i. Therefore, r r fe* fi si* e = fi s = f i=l i=l so V1z = V~L {(E). (iii) b(E)C_ A1). If se~ E), then sc A), so, by Proposition 0.16, dimDsV < m. So dimDs(V/V1) < X and seA1. This means that se 7A1). (iv) V/V1 is finitely generated 1(E)-module. It is an immediate consequence from the assumption that V is a finitely generated NE )-module. (v) Let A1 be the set of all semi-linear transformations on V/V1 induced from A; i.e., \1eA1 is given by Xl(V) = \(v+Vl) = \V + V1 = v for all veV and some AcA. -1 Let G1 = (glgli(a)~ — Mall, ValeAl and some 2leA1]. Evidently, G1 is a group of automorphisms of A1 and [A1:EDT1] < 00 where the T1 is the group of all linear transformations on V/V1 contained in Ai. (vii) V/V1 is not a completely reducible DE-module since n > 2.

29 Hence, by the induction hypothesis, there exists p / 0 in HomD(V/Vn_l, V2/V1) so that pklV =?lpV for all lAEA1 and vEV/Vn_1. Now choose Q with V2 DQDV1 so that the image of V/Vni1 under p is Q/V1. Certainly, Q admits A, it might not admit E. Let P = X aQ. aeE Then P is a DE-submodule of V2 and P/V1 is a completely reducible DEsubmodule of V2/V1 by Proposition 0.5. It is easy to see that P and W/pL are a pair of dual vector spaces with bilinear form given by (v,f) = (v,f) for fc W/P, veP. Now let A2 = (P, W/P'). Since (W/P%) E CW/P1, E can be considered as a subring of A2. Next, we shall again verify the conditions (i) —(v) and (vii) for P and W/P1. (i) )E)P = P. Since 4'(E)V = V, V is a homogeneous completely reducible E-module (by Proposition 0.15), hence, by Proposition 0.5, P is. Applying Proposition 0.15 again, we have'(E)P = P. (ii) (W/P ) f(E) = W/P1. We see that for any feW, there is e = e in (E) so that f = fe. Hence fe = fe = f for all fC W/P and so

3o (W/P1) t( ) = W/pl (iii).'(E)C __A2). If seCE), then set(A), so dimDsV < o, so dimDsP < o. But sEA2, so se A2). (iv) P is finitely generated (E)-module. Since V is completely reducible E-module, V is completely reducible AE)-module. By Proposition 0.4, V = POP' ()ftE)-direct). P is then a finitely generated E)module since V is. (v) Let A2 = (X2 12v = =V VvEP and some AcA}. Then A2 is a group of semi-linear transformationrson P since A admits Q and so admits P. Let _1 Evidently, G2 is= g2roua2p of automorphism2, Vaof A2 and someA2 < o where Evidently, G2 is a group of automorphisms of A2 and [A2' DT2] < 00 where T2 is the group of all linear transformations on P contained in A2. (vii) P is not completely reducible DE-module since P ~ V1. Therefore, we can use the induction hypothesis again to obtain a q 0 O in HomD(P/V1, V1) so that q?2v = \2qcV V?2E.A2, veP/V1. Now since q ~ O and P/V1 = E a(Q/Vj), there existsan aeE such acE

31 that qa(Q/VL) / 0. Consider the mapping gap in HomD(V/Vn-1 V1), qap(V/Vn_-) = qa(Q/Vl) t 0 Also, for any A\c, vcV/Vn-i,, \qapv = a2 qapv = qA2apv = qa\2pv = qa\pv = qa\lpv = qap\lv = qap\v. Therefore, qap may be taken as the desired h.

CHAPTER 3 PROOF OF THE MAIN THEOREM (CONTINUED) In this chapter, we will consider the case n = 2. In this case, V1 and V/V1 are completely reducible DE-modules. Write V = V1@U (D-direct) and let A = EndDV and Av = (aeA aVlCV1 ). It is obvious that AV is a subring of A containing E, and we have maps Q: AV EndDV1 = B v: AV-End U = C 8: AV +Hom D(,V1) = H given by av = Q(a)v au = S(a)u + b(a)u for asAV, ucU, vcV1. Then 0 and ( are ring epimorphisms and 5 is a G-$ derivation, i.e., 5(ab) = Q(a)b(b) + b(a) $(b), a,beAv. We shall show that 0 and $ are one to one maps on E. Let I,= (aeEEl(a) = O). If 0t, 0, then, as an ideal of E,,_~ 2tE). It would follow that t(E)V1 = O, a contradiction. Hence, 32

33 Q,= 0, G is a one to one mapping from E onto 0(E), and hence G(E) is a subring of B. Since B can be considered as a, sub-lgebra of A, and (E)C tA) by Lemma 1.3, t(E) is a finite dimensional algebra over Z. Hence ~~=(BB(G(E)) being a sub-algebra of 14(E), is a finite dimensional algebra over Z. Moreover, since V1 is a completely reducible DE-module by Proposition 0.6, V1 is a completely reducible.-module so )6 is semi-simple. Now, let ~= (aElEj (a) = 0). If 0.. 0, then, as an ideal of E,0_ ~(E ) and this would imply that V = ('E)vQ. VCV1 a contradiction. Hence, 0,= O, so is an isomorphism from E onto p (E). Let $ be the projection on U along V1. Then, for any aEE and vcV, v = vl + u with v1EV1, uEU, $(av) = $(a(vl+u)) = - (au) = $(K (a)u+b(a)u) - (a)u = $ (a)D(v) so U = O(V) = 0( (E)V) = i ( (E))~(V): 4(~ (E))~(v) ='($ (E))U. Now, we claim that U is a finitely generated ~($ (E))-module. To

34 see this, let [v1,...,vm generate V over f(E). Then for any uEU, u = alvl+...+amvm with aictE) and hence u = 0(u) = c(alvl+.. +a vm) - (al)(vl) + +... + (am)(Vm) This means that U is generated by (O(ul),...,$(um)] over' ( $(E)). We assert that V($ -(E))C t(C). Indeed, for any se ~(E), dimDsV < c, hence dimDO(sV) < o. However, O(sV) = B(s)O(V) = ~(s)U, so dimD $(s)U < oo and hence $(s)E(C). Thus ~(/ (E)) =((E) )C t(C). By Lemma 1.3, I= C C(/ (E)) is a finite dimensional algebra over Z. Also, since U, D-isomorphic to V/V1, is a completely reducible D ~(E)-module, by Proposition 0.6. U is a completely reducible - module and hence by Lemma 1.1, is semi-simple. Now, let = End H. We have the mapping B+- given by (bh)u = b(hu) hecH Since HU = V1, the above mapping is an injection. We will think of B as a subring of I. Likewise, we have the mapping C-y given by (ch)u = h(cu), heH, and again this mapping has zero kernel but it switches the multiplica

35 tion so we will think of C0, the reciprocal ring of C, as a subring of(. Now, we want to show that the subring generated by Xand'?~ is isomorphic to 6z'.O It will suffice to show that BZ,.~ =Bf. To see this, we will apply Proposition 0.20. Obviously B is an algebra over Z, RB) is central simple ideal of B, and [B, ~] = 1. Also L(B)z~ = O with z in fwould imply (B)Hz = 0 in H or =B)V1 = 0 contradicting the fact that d-(B)V1 = V1. Hence, t(B)zo = 0 implies zO = O. Moreover, by Proposition 0.19, (bl,...,bm) being a linear independent set in B over Z assures the existence an x in B rB such (xbl,...Yxh)9 is a linear independent subset of ~'(B). Therefore, by Proposition 0.20, BJ&$t = B P~. Since 6and 0 both are semisimple, by Proposition 0.22, Z.,ZX ~o is a quasi-Frobenius ring. Since AV1CV1, we can define the mapping G. ~ and 5 on A as before: 0: A-*+the group of semi-linear transformations on V1,: A+the group of semi-linear transformations on U, 6: A+the set of semi-linear transformations from U to V1, given by \v = 9( )v u = )u + )u for uEU, veV1 and AcA. Clearly, the mapping p (X): h+g(p()h~ ( -l)

is an endomorphism on H. First, we shall prove the following lemma: Lemma 3.1. Under the assumptions of our main theorem with n = 2, suppose that A?,>... are representatives for the left cosets of DT in A. If alP(A1) +... + atP(\t) = 0 (1) in End H with aieBC~ then CXi = O, i = 1,...,t. Proof: Let a,...,at be the automorphisms of D associated with A1,... St respectively. Certainly, we may assume that A\ = 1. Suppose ai are not all zero and I is the group of inner automorphi ms on D. We claim that ai / aj mod I for i $ j. Indeed, if aiCj l(x) = dxd for all xeD with deD then, for all x in D, all v in V, d ij )(xv) = d (Yoiaj (x ) )\ij ( v) - d dxd-l jX\jv = xd- (iXj -1)v and it would imply that d- j T or = dtj for some teT a contradiction. Now let ([j) be a D-basis for V1. Set Si: Vl*V1 given by Si( dkk) = Z ai(dk)k, dkeD and

37 Ti EndDV by Tik = Z Sikj j if Q((i)Sk = Z Sikjj ~~~~~ikj~~j with Sikj in D. Then TiS1 dadk = T j i(dk)~k = i (d k)sikjj k j,k and Q(i) O dkk =, i(dk)Q( i) k = j i(dk)Sikjj k so O(Ai) = TiSi for all i. Let (Ck] be a Z-basis for C and ai = Z (bik)1(Ck)r with bik in B. By the above assertion, we can form the equation (1) as bikSih $(?i )Ck = 0 (2) i,k in H for all hcH, and also we can assume that bll ~ O. Since BbllB is a non-zero ideal in B, we have

38 BbllBD)(B). We then choose bil in ~(B) with bil i 0 and bll = Z bfbllbf, bfbfEB. Multiply the identity (2) by bf from left. We obtain E k bfbikSih/(?i-)Ck = 0. 1 Replacing h then by bfh, we have L bfbikSibfh (-/i )C = 0, i,k Sum over fo We get bikSihk ( i' )Ck = (3) i,k for all h in H, where bik = f ik ibf i in Bo Let eeB given by eil = 1j and ej = 0 for j / 1l Then, obviously, eEY(B) Now we select bll = e and assume that the expression (3) has the smallest number of non-zero biko Then by (3 (l1-e)bikSih$(ij )Ck = 0. i,k

39 in which the first term is (l-e )eSlhi( D1 )Ck = since e e, so (1-e)bik = 0 for all i and k. Now if bikj = Zdikjlle then o = (l-e)biktj = E (l-e)dikj~el = Y dikje(l-e)e~. But (l-e)S~ = 5~ for all I~ 1, he nce dikj = 0 for all I 2 1, ikj~ and bikVlC< 1 > the Dnspace generated by S, for all i and k.

40 We note that Si(l-e) = (l-e)Si for all i, since Si(1- ) k dk k = Si dk k = 4 ai(dk) k and (l-e)Si Z dkk = (l-e) i(dk)k k k = k Jl ri(dk) k. Now in (3) replacing h by (l-e)h, we obtain bikSi(l-e )h(ii- 1)Ck = 0 ik for all h in H, or bik(l-e)Sih(?\i-1)Ck = (4) i,k for all h in H. From (3) and (4), we have by subtraction j [(i-e)bik-bik(l-e )]Sih (ki- )Ck = 0 for all h in H. in which the first term is equal to zero. Hence I? (1-e)bik - bik(l-e) = 0 for all i and k.

41 But we have seen before that (l1-e)bik = 0 so bik(1-e) = 0 or bik = bike for all i and k. Thus, for j f 1, I T bik = bikej = 0 and b ikl = dike for some dik / 0 in D. Now for each d / 0 in D, let Td be the linear transformation given by Td-k = dek In identity (3), replace h by Tdh and then multiply Td-1 from the left in both sides of the obtained identity. We obtain j T _lbikSiTdh (Mi-1)Ck = 0 ( ik for all h in H.

42 Remark that since for all j (SiTSi- 1)j = SiTdt = Sidt = ai(d)t SiTd = Tai(d)Si' From identity (5), we have I Td-lbikTai(d)Sih(?ki1 )Ck = (6) i,k for all hEH. From (3) and (6) by subtraction, we have (T dlbkT i(d)-bik)Sih(=i-1)Ck O i,k for all hcH, in which the first term is T lbllTai(d) - bll = 0, so bik =T TdikTai(d) for all i and k. It follows that!! dik 1 = bikt1 = (T -lbikTai(d)) )1 Td lbikMi (d)1 = Td-lai (d )bik 1 Td _i (d)dikl = ci (d)dikd-1 1,

43 so dik = ii(d)dikdor Oi(d) = dikddik for all i and k. This means ai,l mod I and hence t = 1, so bll = 0 a contradiction. This completes the proof of the lemma. Notice that the lemma generalizes the classical lemma of Dedekind. Now we go back to our proof of the main theorem. Let t F =X P(i) i=l where [k1 o,ht) is again a set of representatives for the cosets of DT in A and An = 1. Then for any i and j, there are deD, teT and a-Ak so that kikj = dtkk. Also) for hcH P(Aij)P(Aj)h = Q(dt\k )h$(Ak-lt-ld-1) = Q(t)Q(\k)h~(0k mt l) = ~(t)p(Ak)h(t)-l) where t commutes with elements of E so @(t)Ek- and ~(t -)e. Hence

44 P(\i )p(\j )E;op (~k )CF. Since (Ai )-i XG1) E and we have 8(Ai)J = ~5 o(A) and O(~ ) = (~ so F is a subring of End H containing, and F is a finitely genera o-module. Now consider the mapping p. from F onto, ~o given by tim im p.1 1 xijyijp(YAi = xljYij 1 j=l j= This mapping is uniquely defined since by Lemma 1.4 each element in F Moreover, for any x in ~~~~~~~~J, y in

im 4J (xyO E E Xijyijp(Ai i=l j=l i 12 = xy Xl. j yl j=l t im = xyl XijYijP(Ai) i=l j=l and there are no one-sided ideals of F other than zero contained in the kernel of 1i. Therefore, [i is a Frobenius homomorphism from F onto;6 o. Thus, F is a Frobenius extension of 0; and by Proposition 0.21, F is quasi -Frobenius ring. Now, we know that [A,E] = 1 and AV1CV1, so [G(A),Q(E)] = 1 and ['(A)(E)] = 1. Also, for \Ae, acE, 5(aX) = 5(Za), or 5(a)/(X) + O(a):(7) = b(i)~(a) + O(A)8(a), or w( )8(a) - 6(a)r(A) = Q(a)8K() - 8(X) (a), where b(a)eHom (U,V1) D~

Let L = (TcF h(rT)cH with 9b5(a) = G(a)h - h$(a),VacE}. If e-X, rE leL, then:~p(Mi)Tr (a) = SP(Ai)T (a)~ - (Xi) [Q(a)h-h~(a) ](AXi-1 ) - G(a)G(%i)hh($i( l ) - QO(Ai)h~(%-l)(a) ) = Q(a) Q(Xi )h~(ki-z) - ) Q(Ai)h~(Ai-1) (a ) so L is a left ideal in F. Furthermore, L ~ F since 1EL would imply the existence of hcH with b(a) = O(a)h - h$(a) for all a in E; i eo, 1E is inner QO-derivation. Consider X = (u-h(u) lucU)CV. X is a D-module and XnV1 = (0) since w = U - h(u)eV1

47 implie s u = h(u) + wcUnV, = (0) so h(u) = 0 and hence w = O. Moreover, for any vcV, v = u+w with u in U, w in V1 and then v = u - h(u) + h(u) + weX + V1. Hence V = X G V1 (D-direct). Now we assert that X is a DE-module. In fact, for acE and uEU, a(u-h(u)) = Q(a)u - Q(a)hu = O(a)u - b(a)u - h$(a)u -= (a)u - h~(a)ucX. This fact contradicts our assumption that V1 is the DE-socle of V. Therefore, IL and L / F. Now since F is quasi-Frobenius ring and L ~ F, the set (xcFILx = 0) 0 9, so there exists f / O in F so that Lf = 0, and so a p in H exists with

48 "h = fp f 0 and Lh = O. However, G(X'-l) [O(X)- ~(X) ]b(a) = (~,-1) [0(x)5s(a)-5( a )(?x) ] = 0(,-1) [0(a)5:(,)-5 (~,)~(a)] = 0(a)[0(,-!)5(,) ] - [0(,-1)5(,) ](a), and _o(-) [o(' )_-O() ] = 1 - p(A-)EF, and so @(x-) [O(,) -o(h) ]EL V\eA. Hence o(x-12) [(x)-~(x)]h 0 V xEA or O(A)h - h$(Q) = 0 V\EA. This h induces an h f 0 in HomD(V/V1,V1) with hkv = khv, VAEA, VcV/V1. This completes the proof.

BIBLIOGRAPHY 1o Artin, E. Galois Theory, Second Edition, Notre Dame (1959). 2. Artin, E., Nesbitt, C., and Thrall, R., Rings with Minimum Condition. University of Michigan (1944). 3. Cartan, H., "Theorie de Galois pour les corps non commutatifs," Ann. Ecole Norm. 64 (1947), 59-77. 4. Dieudonne, J., "Sur le socle d'un anneau et les anneaux simples infinis," Bull. Soc. Math. France, 62 (1942), 46-75. 5 ____"La theorie de Galois des anneaux simples et semisimples," Comment. Math. Helv., 21 (1948), 154-184. 6. Eilenberg, S., and Nakayama, T., "On the dimension of modules and algebras II," Nagoya Math. J. 9 (1955), 1-16. 7. Hochschild, G., "Automorphisms of simple algebras," Trans. Amer. Math. Soc., 69 (1950), 292-301. 8. Jacobson, N., "A note on division rings," Amer. J. Math., 64 (1947), 27-36. 9., "On the theory of primitive rings," Ann. of Math., 48 (1947), 8-21. 10., Lectures in abstract algebra, I, Basic Concept, Van Nostrand, New York (1951). 11o, Lectures in abstract algebra, II, Linear algebra, Van Nostrand, New York (1953.) 12. _, "Structure of rings," Amer. Math. Soc. Colloq. Publ., 37, New York (1956). 13. Jacobson, N., and Richart, C.E., "Jordan Homomorphisms of rings," Trans. Amer. Math. Soc. 69 (1950), 479-502. 14. Kasch, F., "Grundlagen einer Theorie der Frobeniuserweiterungen," Math. Ann. 127 (1954), 453-474. 49

50 BIBLIOGRAPHY (concluded) 15. Nakayama, T., "Galois theory of simple rings," Trans. Amer. Math. Soc. 73 (1952), 276-292. 16. Rosenberg, A., and Zelinsky, D., "Galois theory of continuous transformation rings," Trans. Amer. Math. Soc. 79 (1955), 429-452.

UNIVERSITY OF MICHIGAN 1111111111111111111111111111 251 811117 3 9015 03125 8778