Structure Theorems for Noncommutative Complete Local Rings Citation Fisher, James Louis (1969) Structure Theorems for Noncommutative Complete Local Rings. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BMXA-R647. https://resolver.caltech.edu/CaltechTHESIS:02222016-135018005 Abstract If R is a ring with identity, let N(R) denote the Jacobson radical of R. R is local if R/N(R) is an artinian simple ring and ∩N(R) i = 0. It is known that if R is complete in the N(R)-adic topology then R is equal to (B) n , the full n by n matrix ring over B where E/N(E) is a division ring. The main results of the thesis deal with the structure of such rings B. In fact we have the following. If B is a complete local algebra over F where B/N(B) is a finite dimensional normal extension of F and N(B) is finitely generated as a left ideal by k elements, then there exist automorphisms g i ,...,g k of B/N(B) over F such that B is a homomorphic image of B/N[[x 1 ,…,x k ;g 1 ,…,g k ]] the power series ring over B/N(B) in noncommuting indeterminates x i , where x i b = g i (b)x i for all b ϵ B/N. Another theorem generalizes this result to complete local rings which have suitable commutative subrings. As a corollary of this we have the following. Let B be a complete local ring with B/N(B) a finite field. If N(B) is finitely generated as a left ideal by k elements then there exist automorphisms g 1 ,…,g k of a v-ring V such that B is a homomorphic image of V [[x 1 ,…,x k ;g 1 ,…,g k ]]. In both these results it is essential to know the structure of N(B) as a two sided module over a suitable subring of B. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Dean, Richard A. Thesis Committee: Unknown, Unknown Defense Date: 17 March 1969 Funders: Funding Agency Grant Number Ford Foundation UNSPECIFIED Caltech UNSPECIFIED Record Number: CaltechTHESIS:02222016-135018005 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:02222016-135018005 DOI: 10.7907/BMXA-R647 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9577 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 23 Feb 2016 15:52 Last Modified: 19 Aug 2025 18:15 Thesis Files Preview PDF - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

STRUCTURE THEOREMS FOR NONCCMHUTATIVE COMPLETE LOCAL RINGS Thesis by James Louis Fisher In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Techn~logy Pasadena, California 1969 (Submitted .March 17, 1969) ii ACKNOWLEDGMENTS I wish to thank my advisor, Professor Richard Dean, for his influence in this stage of my mathematical development. His enpouragement, questions and comments have been invaluable in developing and clarifying the material in this thesis, ~nd his suggestions have greatly improved. its exposition. I also wish to thank the Ford f.'.)undation for financial support during my first year of graduate study, and the California Institute of Technology for various teaching assistantships and research assistantships which were granted during the remainder of my stay. iii ABST1-{ACT If R is a ring with identity, let N(R) denote the Jacobson radical of R. H is local if R/N(R) is an artinian simple ring and nN(R)i = o. It is known that if H is complete in the N(R)-adic topology then R is equal to (B) , the full n by n matrix ring over B where E/N(D) n is a division ring. The main results of the thesis deal with the structure of such rings B. In fact we have the following. If B is a complete local algebra over F where B/N(B) is a finite dimensional normal extension of F and N(B) is finitely generated as a left ideal by k elements, then there exist automorphisms g 1 , ••• ,gk of B/N(B) over F such that B is a homomorphic image of B/N[[x 1 , ••• ,xk;e;1 , ... ,gk]J the power series ring over B/N(B) in noncommuting indeterminates xi' where xib = gi(b)xi for all b E B/N. Another theorem generalizes this result to complete local rings which hqve suitable commutative subrings. As a corollary of this we have the following. Let B be a complete local ring with B/N{B) a finite field. fl(B) If is finitely generated as a left ideal by k elements then there exist automorphisms g 1 , ••• ,3k of av-ring V such that B is a homomorphic imase of iv V [[xl ' · · · 'xk; gl ' • · • 'gk]] • In both these results it is essential to lrnow the structure of N(B) as a two sided m0dule over a suitable subring of B. v TABLE OF CONTENTS Page ACKN C)WLEDGMENTS 11 ABSTRACT iii Section I INTRODUCTION 1 II DEFINITIONS AND NOTATI~N 5 III THE COMPLETL:N OF A SEMILOCAL RING WITH FINITELY GENERA'l'ED RADICAL 7 IV COMPLETE LOCAL AND SEMILOCAL ALGEBRAS 14 v COMPLETE LOCAL RINGS WITH SUITABLE COMMUTATIVE SUBRINGS 21 REFERENCES 33 1 S~CTION I INTRODTJC'l'ION 'l'he structure theory of commutative semilocal q,nd local rings has been developed extensivel~ but there has been little success as yet in extending these results to n0ncommutat1ve rings. In this paper we prove some structure theorems for noncommutative complete semilocal and local rings. In Section III we obtain some information on the structure of complete semilocal rings whose radical is finitely generated as a left ideal. In commutative theory it is known that such rings are direct sums of complete local rings. As the ring of triangulqr n by n matrices over a field shows, this result is not true for noncommutative c0mplete semilocal rings but we do obtain the following. (Throughout we shall let N(S) denote the Jacobson radical of the rings.) Th~I.!!.._g_. If R is a complete semi local ring with finitely generated radical, then R = R1EB ..• EBRk where R1 is -:i.n A.lgebra over the rationals and for i > 1, R is a ring with identity 1 e 1 such that for distinct primes p , p e E N( R ). 1 1 1 1 It is well known that the completion of a commutative noetherian local ring is again a noetherian local ring. This problem has not been settled for noncommutative local rings. Goldie [6] shows that this would be a strong con- 2 d1tion. In any left noetherian ring of course the radical is finitely generated as a left ideal. In the case of commutative complete local rings, finite generation is sufficient to prove the structure theorems and to imply that the completion is noetherian. In the nonc(")mmutative case finite generation of the radical is not sufficient to prove that the completion is noetherian as the power series ring F((x,y]J in noncommuting indeterminates x and y over F shows. However, in the noncommutative structure theory, finite generation of the radical seems to play an important role. Theorem 1. In Section III we prove The completion of a semilocal ring with fin- itely generated radical is again a semilocal ring with finitely generated radical. I.S. Cohen [3] proved that any commutative complete local ring with finitely generated radical is a homomor- phic image of a power aeries ring with a suitable coefficient ring. Some effort has been made toward extending these results to noncommutative complete local rings. Bathe [2] for exqmple has shown that if R is a complete noetherian local algebra over a field F such that R/N(R) is finite dimensional and separable over F then R is a homomorphic image of a quasi-cyclic algebra s. This theorem 3 1s extended in Section IV to a complete semilocal algebra R over a field F such that R/N(h) is finite dimensional and separable over F. This theorem is the basis of Section IV and. the structure theorems for complete local algebras. It is known that if R is a complete local ring then R is the ring of all n by n matrices over a complete local ring B where B/N(B) is a division ring. In Section IV we obtain the following information about the structure of B. Theorem 7. Let B be a complete local algebra over F where B/N(B) is a finite dimensional and normal division ring over F. If N(B) is finitely generated by t elements then there exists automorphisms g , ••. ,gt of B/N(B) fixing F 1 such that Bis a homomorphic image of B/N(B) [[x , •.. ,xt; 1 , •.• ,gt]], the power series ring over B/N(B) in noncom1 muting indeterrninates xi where x b gi(b)xi for all b E g 1 = B/N(B). Note that in this theorem we do not say that the automorphisms are distinct. em for comulete local We also obtain a similar theor- algebras B where B/N(B) is a finite dimensional separable extension of F. In Section V we extend the results of Section IV to complete local rings which have suitable commutative sub- 4 rings. '.:)uppose B is a complete local rir .. ; with finitely generated radical and B/N(B) is a field F. Also suppose that the characteristic of B/N(B) is a prime p and the characteristic of B is not equal to p. The major results are these: Theorem 15. Let S be a commutative subring of E which maps onto F under the natural map ~ which takes B onto B/N(B). Let S be a finite module over a subring S' of the center of B, S' be noetherian and F be a normal extension of ~(S'). If N(B) is finitely generated by t elements, then there exists automorphisms g 1 , •.• ,gt of a v-ring V such that B is a homomorphic image of V V [[x 1 , ••• ,xt;g 1 , ••• ,gt]] where V/N(V) is isomorphic to F. Theorem 16. If F = B/N(B) is a finite field and N(B) is finitely generated by t elements, then there exists automorphisms g 1 , ••• ,gt of av-ring V such that Bis a hom0morphic image of V[[x 1 , ••. ,xt;g1 , •.. ,gt]] where V/N(V) is isomorphic to F. 5 SECTION II DEFHHTIJNS AND NOTATION 'I'his section introduces some of the terminology and definitions used in the structure theory of complete noncommutative local rings. All rings will be assumed to have an identity. A ring is artinian if the descending chain cnndition for left ideals of the ring is satisfied. A ring is noeth- erian if the ascending chain condition for left ideals of the ring is satisfied. The Jacobson radical of the ring R will be designated by N(R) or by N whenever the ring R is clear from context. The set of all n by n mat- rices over the ring B will be denoted by (B)n. A semilocal ring R is a ring such that nNi = O and R/N is artinian. A local ring is a semilocal ring such that R/N is simple. A topological ring is a ring which is a topological space such that addition and multiplication are continuous. If I is a.n ideal of the ring R such that n I n = O then R is a topological ring using lrn} as a base of neighborhoods of {o}. The topology is called the I-adic topolosi. A complete ring is a ring which is complete with respect to its N-adic topology. We will designate the c,.,mpletion ,. of R in the N-adic topology by R. A v-ring as introduced by Cohen (3] is a complete discrete commutative valu'.3.tion rine; wh•)se m':lximal ide·:i.l is generated by the prime integer p where p = characteristic of V/N(V). That is, a v-ring is a comr)lete c,,;mmutative local integral domain whose maximal ideal is i::;enerated by the prime integer p. If V is any commutative ring , e; , ••• ,g1c: automorphisms 1 of v then the Hilbert power series ring V[[x , .•• ,xk;g , ..• 1 1 ,gk]] is defined as the power series ring over V in noncommuting indeterminates x 1 , ••• ,xk such that xiv= gi(v)xi for all v f V. An algebra R is ~·1sicyclic if 1) R C'.'ntains a sub- algebra A which is mapped is·1morphically onto H/N bv the natural map, ~nd 2) as a two sided A module R is equal to the complete direct sum A$ N $Ni$ • • • 1 where N ~ 1~/N 2 • Let R be semisimple artinian algebra over F. 1 Let the Vledderburn decomposition be ( D1 ) n ( 1 ) $ ••• EB ( D1::) n (l{) • Let Ci be the centers of the division rings Di. Such an algebra R is seuarable over F if each of the fields Ci is a separable extension of F. A separable algebra is normal over F if each Ci is a normal extension of F and each automorphism of Ci over Fis induced 1y an autom~r­ phism of Di over F. Finally we next introduce the notion used by Hochschild [8] of a U-regular (V,V) module where V is a ring ~nd U is a subring of V. All-regular- (V,V) module M is a two sided V module with U c {v E V : vm = mv for all m E i!; l· 7 SECTION III THE C ,;.,;_ Ll!.'l'I Ol~ >F' A 3&.;ILOCi~L RH;G '.TI'H FINI TJ.<.;LY (}Ui;ERA'l'EI.J RADICAL This section gives us s-ime general 1nformati'in on the completion of a semilocal ring whose radical is finitely generated as a left ideal. We relate semilocal rings with finitely generated radical and projective limits of artinian rings in order to obtain further inf0rmation. The followins lemma is easily derived from theorems 1.3 and 2.3 of [l]. If R is a ring with identity such that nN 1 = c, Lemma 1. then nN(R)i = O, while for j > O, l'-T(R)jn R R/N ( R) .1 ~ = l•i(ri)J and R/H (R) j • We will now investigate some properties of finite generation of the radical Lemma 2. If R is 0 ,q_ complete semilocal ring with R/N"' artinian then for each positive integer j, Nj is a finitely generated left ideal. ~roof. We will first show that N is a finitely left ideal and use this to obtain the lemma. ~. le1 t ' .,.- R/~~ gener~ted Let n E 1-~. . module we h8.ve n = ~k r n a minimal generating set ~=l r~r the i i k 2 R./0i rr.odule N/N • where t,s n- Lri n 1 Since t,s N- N' there exists E ! and { s } 1 1 such that t- t t i n 1 1 L ~1 because n. s 1 2 N- N E 1 I~ 2 but m+r~3 = L:ts+N3 '? ~lft1te I l k f Lei 2 N- N • f =m E N?, l ti lt=l s- t s ni E N'.) so 1 i. •\ and s0 is written Continuing this process we have E Nu+l. Since R is complete we have Now we rr:q,y reo:i.rra.nge the series and collect terms ending in nq• Thus we haven= u 1 n 1+ •.• +uknk where for example (1) k (j) ul = rl + L.Lri(l) ... i(j-1) ,1 ni(l) · · .ni(j-1) · j ~l. <1111 ·,iC.i"') ., 00 Therefare N = [n 1 , •.. ,n~]. We will now show that the set of all products ni(l)•••ni(j) generate Nj. 1he statement = 1 so by induction it is sufficient to is true for j show that the statement is true for q+l if it is true for q Let n · E Nq+l (u) N. • Therefore n (u) q =~ s ( u) s (u) where s (u) LI 1 · · • q+l i """' Since s 2 •.• sq+l EN , we have 9 (u) ~1.(u) (u) 2 · • • 8 q+1 =. ~ r 1 ( 2) . . • i ( q + 1 ) n i ( 2 ) · · • n i ( q + 1 ) and t .. LISl,···,•'f"' t n = ~s(u) LI \l.• I 1 .. r(u) \.Ill,.... •"\• I) "'' 1(2) ... i(q+l) n 1(2) 000 ni(q+l) E 9 i:. II. = L ( L 9 (~);,i(b) ... i(q+l) )ni(2) · · .ni(q+l) ~ti.1.).1·· • .,'4rftt\ ., Eut "'- : I ~ L:s(~) ~i(b) ... i(q+l) f N and hence is equal to .... , " ""r n £..J i(l) ••• i(q+l) i(l). Therefore i..l1\Z' l I< L n =it1i,...,,,,.,,ri(l) •.• i(q+l) ni(l) • · .ni(q+l) and the lemma is proven. Note that in proving this lemma we showed that any minimal generating set of N/N? as an R/N module can be raised to a minimal generating set of N as an ideal and vice versa • Hence from the properties of finitely gener- ated modules over semisirnple artinian rings we know that every minimal generating set of N has the same length. Theorem 1. The completion of a semilocal ring with fin- itely generated radical is again a semilocal ring with finitely generated radical. Proof. Since N is a finitely generated left ideal it 2 follows that N/N is a finitely generated left R/N module. 2 Since R/N is artinian by assumption it follows that R/N is artinian. " " 2 By lemma 1, R/N(R) is isomorphic to R/N{R) 2 and is therefore artinian. lary. Lemma 2 then proves the coral- 10 i~r0posi tion 1. If R is g complete sernilocal ring with R/N 2 artinian then for each positive integer i, H/Ni is artinian. Proof. By lemma 2 for any k >O, Nk is a finitely generated ideal. Hence Nk/Nk+l is a finitely generated left R/N module. 'rhus a finite composition series can be obtained from the series R/N, N/N 2 , ••. ,Ni-l/Ni. This implies that R/Ni is artinian. Some information about the structure of complete semilocal rings can now be gained from our knowledge of artinian rings ty using the projective limit. Suppose that { Ri}, (i ~ 1) is a sequence of rings and {cJ>i}, (i ~ 1) is a sequence of homomorphisms such that cJ> 1 _1 (R1 )= Ri-l" The projective limit of {R1 } is defined to be the subring of the complete direct sum of {Ri}' consisting of those elements {ri} which satisfy cJ>i_ 1 (ri} = ri-l" The projective limit of IRil will be denoted by PL(Ri). The following is a well known and easily proven fact. Proposition 2. If R is any ring complete in the N-adic i topology where n N = 0 then R is the projective limit of : RI N - R/N lR/N II under the natural map "'i-l 2 Proof. i ,I. The map r E i i-1 • R - (r+N, r+N , ••• ) is a horrwmorphism 1 from R into PL(R/N ). It is 1-1 since r _. 0 implies 11 r Ni for each i which in turn means r f pose (r +N, r +N 2 , .•. ) 1 2 k rli. = ~(ri-ri-l) f PL(R/Ni). f n Ni = o. Letting r is a Cauchy seq~ence so that r = 01 But r-r = ~(r -r ) u j ...... , j j-1 ) (r+N, r+N'-, ••• ) = (r +N, r +N2 ·, •••• 2 1 Theorem 2. o we have 0 ~(ri-ri_ ) exists. ~ = Sup- E Nu so that Let R be a complete semilocal ring with finitely generated radical. Then R = !\EB· •• e:Ri{ where R is an algebra over the rationals and for i > 1, R is a rin13 with 1 1dent1 ty e 1 such that for distinct primes pi, pi ei Proof. E N (R ). 1 ~e will make use of the fact that R = PL(R/Nj). R/N is artinian and hence is equal to R11e ••• eRkl where R11 is an algebra over the rationals and for i > l, Ril has characteristic pi for distinct prime integers pi. Since 1 E R ?..no R/NJ is artinian ( Pr·;posi ti on 1) there exist by [5, page 283] rings Rij , 1 s is k, 1 s j R/Nj such that = I\jf'.B ••• ef~Ij where I\. is an algebra over the rati-::;n1 als and for i > 1, Rij has characteristic a power of pi. Because of the distinctness of the characteristics of the c-:>rnponents of the decomposition of R/Nj, it is easy to see that this decomposition is unique. deduce that the natural map </>j map 1 j : Ri,j+l-+Rij" <1> we have N/Nj+l From this fact we can : RI Nj +l --+ R'/N·1 induces a From the decomposition of R/!·~j+l = N(Rlj+l)E& ..• E&N'(2kj+l) 'lnd 1)/Nj+l= N(Rlj+l)je ••• eN(Rkj+l)j. Thus ~j is actually the natural 12 Frnm this it is evident map of Rij+l nntn R1 .1+1 /N(Rij+l)j that the diagra.;r, h/N.1 /NJ + l .!.L : l ulj <Pij l uij+l Rlj - ?li.i+l commutes where uij is the projection map of R/N,1 onto Rij. We now claim that R = PL(R/Nj) =PLP'l.l )EB •.• EBPL(Ekj). The commutivity of the diagram guarantees that u: {ri+Ni} --.{ r 1 i, .•• ,r1{1 j is a well defined map. It is easily checked that u is an isomorphism into. Suppose that lr 1 .i} E PL(Rlj). sequence jri+Nil 'ro show that u is onto we must have a f PL(R/Ni) which maps rmto lr1 ,,,o, ... ,oj under u. The seq_uence lr1 .1+Njl of course does this. We now need to show that R1 PL(R1 J) contains a copy of = the rationals and that N(Ri) = N(PL(Rij)) c(":ntains p 1ei• To show the first note that if R11 is zerr:· then R1 .1 is zero for every j s.nd of course PL(R1 j) is zero. Thus R1 is trivially an algebra over the rationals. Otherwise R1 J contains a unique copy of the rationals generated by e 1 .1 the identity of Rlj• <Plj-l maps this copy onto the unique copv in R1 ~-l so that PL(R1 j) is an algebra over the r~tionals. The assertion that N(~i) = N(PL(Rij) con- tains piei is clear since piei is contained in :N(Rij) and <Pij-1 maps pieij onto pieij-l• The theorem is thus proven. 13 Corollary 1. :i?or i > 1, either Ri ht:ts chg,r:1cteristic a power of a prime or Ri contains a copy of the valuation ring Ip(i) determined by the prime pi in the integers. Proof. R1 = l)L(R1 /N(Ri)j) and each R1 /N(Hi)J contalns Iei/N(Ri)jwhere I is the inte3ers and e 1 the identity c,f R . 1 If there is some integer n such that piei f n each j then piei p~. = 0 and R h~s N(Ri)j for characteristic dividing If this is not the case then the localization and completion of the subring PL(Iei/N(Ri)j) of R1 ~bout the ideal generated by piei is aeain a subring of Ri and is in fact a copy of Ip(i). Corollary 2. Let Ri/H(Ri) = (D1 )n(l/l1 •.• E1'(Dt)n(t) be the \;edderburn d ecomposi ti on of the semi simple ring Ri/H (Ri) where ej - ej is the map nf ortho5onal idempotents of R1 onto orthogonal idemp0tents of Ri/l,l (R1 ). euRiej. Let Muj = If we mqke the convention thqt a characteristic which is equal to zero is written as equal to infinity then the characteristic of Muj divides min {characterisLic of eu, characteristic of ejl. Proof. Muj is an (eufiieu, ejRiej) module. Hence the characteristic of Muj divides the characteristic of eu and the chara,~teristic of ej. 14 SECTI~)N IV Crn1PL.b.:T.b.: LOCAL ASD 31·1-iILOCAL AL3EBHAf3 This section deals with the structure of certain complete semilocal and local alzebras with finitely generated radical. The techniques used are an extension of some ideas of Hochschild [7 ,9]. ·1·re prove that if R is a complete local algebra over F with finitely generated radical such that R/N is finite dimensional and normal over F then R is a complete matrix r1n5 over the homomorphic image of a Hilbert power series ring. A new type of complete local ring is then defined and we investigate the structure of complete local and separable algebras in relation to this ring. The following theorem was proven by Curtis (4, page 80]. Theorem 3. Let R be an algebra over a field F where n Ni = O and R is complete in the N-adic topology. R/N is finite dimersional and separ~ble If over F then R contains a subalgebra A. such that R = A+ N and An N = O. The next lemma and theorem are found. in Hochschild [7, pages 371 and 372]. Lemma 3. If A is a separable algebra over F then every two sided A module M is semisimnle, in the sense that every two sided submodule of M has a cornple~ent. 15 Theorem 4. Let R be an algebra over F with R/N finite dimensional and separable over F. If Nk = O for some k then R is a homomorphic image of (R/N)e:t-\e ••• eN~-l where ? Ni ~ N/N·-. We now extend these results to complete semilocal algebras. Batho [2] proved the following theorem but we include a proof here using different ter:ninnlogy and concepts. Theorem 5. F. Let R be a complete semilocal algebra over If R/N is finite dimensional and separable over F then R is a homomorphic image of a quasicyclic algebra R/N EB N1 e Nie • • • • Proof. Proposition 2 gives R = PL(R/N1 ). there is a subalgebra A such that R By theorem 3 = A EB N as two sided A modules. Since A is a separable algebra lemma 3 guaran2 2 tees that N has a complement N1 1n N. That is N N1 EB N as two sided A modules. Using this N and A, theorem 4 1 i then implies that R/N is a homomorphic image of Ci = - -i-1 where -A and N are the appropriate N1 EB ••• EBN A EBN 1 e -2 1 1 1 images of A and N1 in R/N • Let cri be the homomorphism 1 of Ci onto R/N • Let <P 1 be the natural map from R/Ni+l onto R/Ni and ~i the natural map from Ci+l to Ci. Note that = the diagram 16 I/Ji Ci+l - Ci 1 cri+l </>. 1cri commutes. R/Ni+l __l R/Ni Let (c 1 ,c 2 , .•• ) E PL{Ci). Define 'Y(c 1 ,c 2 , •.• ) = (u1 (c 1 ), u2 (c 2 ), ••• ) • The commutivity of the diagram implies that i this is a map of PL( Ci) onto PL(R/N ) = R. that 'Y is a h -momorphism. It is clear Note that PL(Ci) as a two sided A module complete direct sum. In order to gain additional information about complete local algebras, we need to know the structure of N/N a two sided R/N module. 2 as The following theorem of Hoch- schild [8, page 451] gives us this structure for certain rings R/N. We restate it for our special case. Theorem 6. Let D be a finite dimensional normal division ring over the field F. Then every K-regular (D,D) space is a sum of simple K-regular spaces. Every simple K-regular (D,D) space has the form Dn where nd = g(d)n, d E D, and g is a fixed automorphism of D over F. If Risa-complete local ring then R = (B)n where B is a complete local ring with B/N{B) a division ring. Hence the structure of R is determined by the structure of B. 17 Theorem 7. Let B be a comnlete local algebra. over F where B/N is a finite dimensional and normal division ring over F. If N is finitely generated then there exist automor- phisms g1 , ••• ,gk of B/N such that Bis a homomorphic image of B/N [[x1 , ••• ,xk;g1 , ••• ,gk]], where l{ is the dimension 2 of N/N as a left B/N sp~ce. Proof. By theorem 5, B is a homomorpnic image of 2 A$ N1 $ N~ $ . . . , N1 = N/N • Theorem 6 tells us that N1 = (B/N)n 1 $ ••• $ (B/N)nk where nib = g1 (b)ni for b E B/N and automorphisms e;1 of B/N. Hence B is a homomorphic image of B/N[[x1 , •.• ,xk;g1 , ••. ,gk]]. Note that k is the 2 dimension of N/N as a B/N module and hence by the proof of lemma 2 n , •.• ,nk is a generating set of N. 1 This set is of course minimal. We now consider the case where B/N is a field and is a finite dimensional separable extension of F. in theorem 7 As we need to gain some information "'n the structure 0f N/N 2 as a two sided B/N module. Since B/N is a separable extension of B we have B/N = F(e) where e is a r')ot of a separ9.ble and irreducible polynomial f (x) contained in F (x]. We now collect the following inform- ation from Jacobson [lo]. Theorem 8. in F(x]. Let f (x) be a separable irreducible polynomial Let e be a root of f(x) and let K = F(e). If 18 M is an F-regular (K,~) module which has finite di~ension as a left K space then M = M1 $ ••• EB Mt as simple birnodules. With each Mi is associated an irreducible factor m(i)-1 m(i) [ -a0 i-alix- •. ·-Bm(i)-l,ix +x of f(x) in K x] such that Mi= Kx 1 iEB ••• EBK~(i)i as left K spaces. multiplication on the right by In Mi e is identical with the linear transformation induced on Mi as a left K space by the companion matrix 0 0 0 1 0 • 8.oi ali 0 ••• 1 •.• 0 0 . 1 8m(i)-l,i m(i)-1 m(i) of -ao1-alix- •• ·-3m(i)-l,i x +x • ;;:roof. Multiplication on the right of M by e induces a K linear transformation T on the left K space H. e satisfies f(x) € F[x] we have f (T) = o. Since Hence the minimal polynomial for T in K[x] divides f(x) and is thus separable. The rational canonical form for linear transformations now guarantees that M = !'~l 9 ••• EB Mt where Mi is invariant and cyclic under T. Thus each Mi is a simple (K,K) module. There is a basis for Mi such that the transformation Ti induced on Mi by T takes the form of the companion matrix for some irreducible factor of f(x) in K[x]. We are now in a position to prove the final theorem of the section. First we shall define a class of co~plete 19 local algebras. Let K = F(8) be a finite dimensional separable extension of F. Let f(x) be the irreducible manic polynomial of e over F(x]. Let T be the companion matrix of an irreducible monic factor of f{x) in K[x]. We may suppose that this factor has degree n(i) so that Ti is an n(i) by n{i) matrix. Let us note here that the Ti are not necessarily distinct and do not neaessarily include all the companion matrices of the irreducible fact•rs of f(x). The ring K([x1 , .•• ,~;T 1 , ••• ,Tt]J is defined to be the power series ring with coefficients from Kin n{l) + ••• +n(k) noncommuting indeterminates x1 j ! where Xi_ = xil' ••• ,xin(i)} , xij f = fxij for all f f F and xije = T1 x1 j where Ti is considered as a linear trans- formation on the space Kxil $ ••• $Kxin(i). Theorem 9. If B is a complete local algebra over F whose radical is finitely generated, and if B/N = K is a. finite dimensional separable extension field of F then there exists matrices Ti such that B is a homomorphic image of K [(x1 , ••• ,Xt;T1 , .•• ,Tt]] where n(l) + •.• + n(t) 2 dimension of N/N as a left B/N space. Proof. = k, the Since K is finite dimensional and separable e:-'.:ten- sion of F we have K = F( e). By theorem 5 B is a homomorphic , the complete direct sum. theorem 8 we have N = M1 $ ••• $Mt where ?~ 1 = Knil $ ••• e Knin(i) and n 1 je = T1n 1 j for the appropriate companion By 20 matrix Ti. Hence Bis a hom0morphic image of K [[x1 , ••• ,Xt;T 1 , .•• ,Tt]] under the map induced by xij---+nij. We may note here that theorem 7 is a direct result of this theorem when B/N is a field but does not follow if B/N is a division ring. Corollary 3. If B is a complete local ring whose radical is generated by k elements, k minimal, and if B/N = K is a finite dimensional separable extension field of the rationals then there exist matrices Ti such that B is a homomorphic image of K [[x1 , ••• , ~; T1 , ••• , Tt]] where n ( 1) + ••• + n ( t) Proof. = k. Since the characteristic of B/N is zero, B is an algebra over the rationals. theorem 9 are satisfied. Hence the hypothesis of 21 SECTION V COMPLE'rE LOCAL RINGS WITH SUITABLE C·J:'i:MUTATIVE SUDRINGS This section deals with complete local rings B whose radical is finitely generated as a left ideal and which have suitable commutative subrings. We will see that these rings include those complete local rings B with finitely generated radical such that B/N is a field F which is a finite dimensional normal extension of its prime subfield. If the characteristic of B is equal to the character1 stic of B/N then the results of this section reduce to the results of section IV. Therefore in the following we will assume that the characteristic of B/N is a prime p and the characteristic of B is not equal to p. The structure of certain U-regular (V,V) modules will be essential in the theorems of this chapter. We use some elementary techniques to obtain the required information in the next series of lemmas. In these lemmas we will assume that B and U are v-rings and V is a finite module over U. 'l'he following lemma is an easy variation of a result of Cohen [3, page 68]. In the version as stated below, we do not need the hypothesis that R is noetherian. Lemma 4. RCS Let R and S be commutative local rings with and R complete. If S·N{R) = N(S) and S/N(S) is 22 a finite al6ebraic extension of R/N(R) then S is complete and S = Ra1 + ••• + Rak where a 1 , .•• , ak is any lifting of a basis a 1 , ..• ,ak of S/N(S) over R/N(R). In the remainder of this section we will denote V/N(V) by F and U/N(U) by F1 • ients 0f V by Lemma. 5. Q We will denote the field of quot- and the field of quotients of U by Q1 • If a 1 , ••• ,ak is a basis for F over F1 and. a.1, •.• 'ak if a set of elements of v mapping onto al' ••• , al{ under the natural ~ap then a.1 , ••. ,ak is both a basis for V over U and Q. over Q1 • Proof. = Ua 1 + •.• + Uak. Lemma 4 guarantees V now show independence over u. pendent over U. T \·/e must Suppose a 1 , .•• ,ak are de- Without loss of generality we may assume = [ u # O : u E U, 0 = ua 1 + ••• + ukak} is not empty. Let S = { a : a is a non-negative integer such that for some u E T we have u for any u" E U}. = pa u I where u I E U and u -;/: pa+ 1 u II Since 'r is not em9ty then under the na t- ural map we have 0 = ual + ..• + vuak which implies u and u = pu' so that S is not empty. 0 Pick a minimal a E S. = Therefore there exists u 1 , ..• ,uk such that 0 u 1 a 1 + •.. +ukak a a+l and u 1 is divisible by p but not by p • In F we have 0 ui = u 1 a. 1 + ••• + ukak so that u1 = 0 for each i. E N(U) which implies u 1 = pu:i_ for each 1. Therefore Theretore 0 =p (u:l. a 1 + ..• + u1~alc) and since V is an integral domain 23 a We have u 1, = pa-1 u , an.d u{'f Pu" for any u" € u. Since a is minimal, we have a= O, which contradicts the fact that u 1 is divisible by p. Hence = 0 and a 1 , ••• ,ak are linearly independent over U. Since Q = [v/pa : v E V, a is a non negative integer} each ui we have v/pa = (u 1 /pa)a1 + ••• + (~/pa)8it so that Q = ~ a 1 + ••• + ~ ak. A linear dependency of the ai in Q over ~ would imply a linear dependency in V over U. Therefore [a1 , .•• ,ak} is indeed a basis for Q over Q1 • Lemma 6. If F is a normal extension of F1 then there is an isomorphism between the group of automorphisms of V fixing U and the group of automorphisms of F fixing F1 • If g is an automorphism of V then an isomorphism is given by g--+g where g(v) = g\V). Proof. Since F is normal over F1 , F is the splitting field of a separable irreducible polynomial r(x) pose r(x) = (x-a1 ) ••• (x-ak) in F(x]. E F1[x]. We may suppose that [ a1, ••• , ak} forms a normal basis for F over F1 • f (x) € u[x] which maps onto f(x). Sup- Choose B,r Hensel's lt-,,mma •' [a1 , .•• ,ak} can be raised to [a 1 , ..• ,ak) contained in V such that f(x) splits in v[x] to f(x) = (x-a1 ) ••• (x-ak). By lemma 5, [a1 , ••• ,ak} is a basis for V over U and Q over Q1 • This implies that Q. = Q1 a 1 + ••• + Q1 ak is the splitting field of the irreducible polynomial f (x) in 24 ~[x], which proves that Q is a normal extension of~· If g ls an automorphism of Q over ~ then g(ai) = an(i) where n ls a permutation of l, ••• ,k. any automorphism of Q over of V over u. ~ = V and Hence g(V) restricts to an automorphism Note that the map g~g = gfVT where g(v) ls a homomorphism of the Galois group of V over U into the Galois group of F over F1 • Since g induces a permut- on a.1 , ••• ,ak. g will induce the same permutation Hence g J: 1 implies g J 1 and g~g is an isomorphism. To see that the isomorphism is onto we note ation of ~, •.• ,81.r, that the Galois group G of V over U has order k as does the Galois group of F over F1 • We are now in a position to investigate certain Uregular (V,V) modules. We will require that F is normal over F , and under this assumption we have 1 Theorem 10. Let M be a U-regular (V,V) module. g , ••• ,gk be the automorphisms of V over U. 1 Let Then Mis equal to M1 e •.• e Mk as (V, V) submodules where mv = gi (v)m for all m E Mi. Proof. Since F is normal over F1 , there is as in the proof of lemma 6 an irreducible polynomial f (x) E u[x] which splits in V[x] and which has a 1 as a root where V = U(a1 ). Suppose f(x) = (x-a 1 ) ••. (x-ak) in V[x]. 25 Define fi{x) = ll.(x-aj). Since Vis local and ai-aj f t N(V). Therefore (f (ai))-l 1 exists in V and we have the identity 1 = I:(fi(ai))-lfi(x) N(V) for i i j we have fi(ai) in V[x]. vie may now note that mul tiplicat1on on the right of M by a 1 is a V linear transformation T on M as a left V apace. Since f (a1 ) = O we also have f {T) = o. of the identity in V(x] we have I Because = I: ( f i ( ai ) ) -1 f i ( T) so that M = I: ( f i ( ai ) ) -1 f i ( T )M = Ml+ • • • +Mk where -1 Mi = ( f i ( a 1 ) ) f i ( T) M. To show that this sum is direct suppose for example that m e: M1 n (M 2 + ••• + Mk) • Since m e: M1 we have (T-a1 I)m = O, and since m e: M2 + •.• +Mk we have (T-a2 I) .•• (T-akI)m = 0. This implies that m Im = I:(fi (ai) )-lfi (T)m = 0 and the sum is direct. Now for every m e: Mi we have since f(T) = O, (T-ai I)m = o. But Tm= ma1 • Since ai is a root Therefore Tm =aim. of f(x), as is a 1 , there is an automorphism gi of V over U which maps a 1 onto ai. mv = m( ~ujai) = t:ujmai Therefore ma 1 = gi(a1 )m and = ~ujgf (a 1 )m = gi ( ~ujai)m = gi (v)m, and the theorem is proven. Lemma 7. Let S be a subring of a field F. If F = Sa 1 + ••• + Sak then S is a subfield of F. Proof. Let Q be the field of quotients of s. Since F = Qa 1 + ••• + Q.ak there is a linearly independent subset b 1 , •.. ,bm over Q with b 1 = 1 and (b 2 , •.. ,bm} c (a 1 , .•• ,ak} 26 such that F = Qb 1 + ••• + Q.bm. qlj Pick s Q,. E 1, .•• ,k ,j ~ si ( t: E S such that sqij E S for every i = = 1, ••• ,m. qij bj). Hence a. 1 = ~ qlj b j where f E F implies f Therefore sf = = r:siai = .. t: si ( f sqi.i bJ) = ~ s.1bj s. This implies F = sF c Sb 1 + ••• + Sbm. For -1 t ES we have t = s 1 + s 2b 2 + •• -tsmbm' and since t- 1 E Q where sj E we have t -1 = s 1 which implies that S is a field. This lemma tells us tha.t if </> is a homomorphism of R onto a field F and R = R'a 1 + ••• + R'ak where R' is a subring of R then </>(R') is a subfield of </>{R). In proving the main theorems of this section we will need to know the existence of homomorphic images of v-rings within our noncommutative local rings. The existence of thee·e v-rings is determined by using the following theorem of Cohen [3, page 79]. Theorem 11. Let R be a commutat1.ve complete noetherian local ring with residue field R/N of characteristic p. Let ~ be the natural map from R onto R/N. Then R contains a subring S which is a homomorphic image of a v-ring V where if 6 is that homomorphism, ~(o(V)) = R/N. We will now investigate the structure of a local ring B of characteristic not eQual to p where E/N is a 27 field F of characteristic p. Let C denote the center of B, <P the natural map of B onto F' and suppose that S is a commutative subring of B which maps onto F under </J. Theorem 12. Let S be a local ring and let S' be a complete noetherian local ring contained in C ns. If s is a finite module over S' and F is a normal extension of </J( S') = F1 then there exist automorphisms, g 1 , .•. ,gk, of av-ring V such that B is a homomorphic image of Ve N as rings where N = N(B). niv N = N1 e ••• eNk as (V,V) submodules where = gi(v)ni for all n 1 E Ni. In proving this theorem we need the existence of v-rings V ~ U and a homomorphism 6 which maps V into S such that o(U) cs', </J(o(V)) = F and </J(o(U)) = F1 • This will enable us to determine the structure of N(B) as a U-regular (V,V) module. We first need the following lemma found in Curtis (4, theorem 2]. Lemma 8. Let R' be a finite extension of a noetherian semilocal ring R. Then R' is a noetherian semilocal ring. If R is complete in the N(R)-a.dic topology then R' is complete in the N(R')-adic topology. Lemma 9. With hypothesis as in theorem 12, there exist v-rings V ~ U and a homomorphism 6 : V~s such that 28 o(U) c S ' , <b ( d ( V) ) = F and <b ( cS ( U) ) = F1 • Proof. By applying theorem 11 to the complete noetherian local. ring s• we obtain a v-ring U and a homomorphism o: U--+S 1 such that <b(o(U)) = F1 • Since F is finite dimensional and normal over F1 we have F = F1 (e) where is a root of an irreducible polynomial f(x) e F1 [x]. Let f(x) E o(U)[x] be a polynomial which maps onto f(x) under the map induced on o(U)[x] by <b. E Since Sis a com- plete noetherian local ring Hensel's lemma guarantees that there is a root e € s such that f(e) = 0 and <b(e) = e. The ring o(U)(e) maps onto F. Let g(x) E u[x] map onto f (x) under the map induced by o. 6 extends naturally to the homomorphism Y which takes U[x]/(g(x)) = V onto o(U)(8). Since g(x) is irreducible, Vis an integral domain. The fact that V is finitely generated over U then implies by lemma 8 that V is a complete local ring. Since V/pV = F we have N(V) = p V. Therefore V :::> U are the appropriate v-rings. Proof of Theorem 12. Define B 1 = VeN(B) where vnw = Y(v)nY(w) for v, w E V, n from V into s. E N(B) ~nd Y is the homomorphism Since Y(U) c C we have (v E V : vn = nv for all n E B'} :::> u. Therefore N(B) is a U-regular (V,V) module and B' is a ring which is mapped. onto B by the homomorphism y' (v + n) = y(v) + n for v E V and n E N(B). 29 By theorem 10 there exist automorphisms g1 , .•. ,gk of V over U such that N(B) = N1 + ••• + Nk as U-regular (V, V) submodules where nv = gi(v)n for all,n E Ni. This completes the proof of theorem 12. Note that in theorem 12, the ring B had certain complete subrings but B itself was not required to be complete nor was its radical required to be finitely generated. If these additional assumptions on B are made we have the following Theorem 13. Let B be a complete local ring whose radical is finitely generated by t elements. hypothesis of theorem 12 If B satisfies the then there exist automorphisms g , •.• ,gt of av-ring V such that Bis a homomorphic image 1 of V[[x 1 , ••. ,xt;g1 , ••• ,gt]] where V/N(V) is isomorphic to B/N(B) Proof. = F. By theorem 12 B is a homomorphic image of B' = VE& N(B), where N(B) = N1 EB ••• E9 Nk. Label elements in 2 N(B)/N(B) with a bar. Hence N(B) = N1 EB •.. e Nk. N(B) is a vector space over F so we may pick a basis [n 1 j} such that [n 1 1, ... ,nij(i)} spans Ni. of Ni which map onto nij" Let n 1 j be elements Hence nijv bering the n 1 j as n 1 , ••• ,nr and the in general will not be distinct) = gl(v)nij" Renum- gl as g1 , •.• ,gr (these we have ni v = e; 1 (v)n 1 • From the proof of lemma 2 [n1 , .•• ,nr} is a minimal gener- 30 ating set f0r N(B). Therefore B' is a homomorphic image of V((x 1 , ••• ,xr;gl' ••• ,gr]]. This in turn implies that Bis a homomorphic image of V((x 1 , ••• ,xr;g , ••. ,gr]]. 1 We of course have that r ~ t and so B is also a homomorphic image of v[[x1,···,xt;g1,···,gt]] where gr+l' ••• ,gt may be any homomorphisms of V over U. The condition that B has a complete commutative local subring S mapping onto F with 8 1 a complete noetherian local subring in the center such that S is a finite module over S' appears to be very restrictive. Theorem 14 will show that if B is complete we can omit the hypothesis that S and S' are complete and local. Theorem 14. : Let B be a complete local ring. Let S be a commutative subring of B such that ¢(S) is the field B/1': where q, is the natural map. If s 1 is a noetherian ring contained in the center C and S is a finite module over s1 then there exists a commutative local subring S' of B which is a. finite module over a complete com~utative local ring S:l contained in C and where t/>(S') = F ¢(3]_) l/>(S 1 ). Proof. Lemma 7 and the fact that S = s 1 a 1 + •.• + s1 ak means that ¢(s1 ) is a field. Since ~11 elements 0f E outside N(B) are invertible we can localize in s about 1 31 the ideal N(B)ns 1 • We may now complete in the radical topology (which is identical to the topology induced by Si· the powers of N(B)) to obtain a ring Note that S{ is contained in the center C since multiplication is continuous in the N(B)-adic topology. ~ u1e mo ..... s•1 a 1 + ••• + s'l l':lk over ~· ~ • 1 Define 3 1 to be the Note that S' is ~ctu~lly a ring since aiaj is contained in s•. By lemma 8 S' is a Since the only com~;:lete noetherian semilocal ring. idempotents in B are O and 1 this means that s' is in fact local. Combining theorem 13 and 14 we obtain Theorem 15. Let E satisfy the hypothesis of theorem 14. If Fis finite dimensional and normal over c/>(S 1 ), and if N(B) is finitely generated by t elements, then B is a homomorphic image of V[[x 1 , ••• ,xt;g 1 , ••. ,3t]] where Vis a v-ring mapping onto F and gi are automorphisms of v. If B/N is a finite field we can drop the assumptions about the existence of Theorem 16. s. Let B be a complete local ring with B/N a finite field F. If N(B) is generated by t elements then there exist automorphisms g 1 , ••. ,gt of av-ring V such that Bis a homomorphic image of V[[x 1 , ••• ,xt;s 1 , .•• ,gt]] where V/N ( V) = F. 32 Proof. The identity of B generates a subring s center of B. in C the Since F is finite F = ZP(e) where ZP is the integers mod p. Let B be a representative of ~ in B. The subring s[e] is commutative and noetherian. In 3(0) localize and complete about s[e] n N( L) to obtain S' is of course a c ·mplete local commutative ring which maps onto F. Ly theorem 11 there is a v-ring V and a homomorphism o such that o(V) c s' anc1 cf>(o(V)) = F. Since o(V) contains the identity it also contains 3 and the localization and completion 3 11 of S about N(B). of course inc and N(S 11 ) = pS". Since H(o(V)) pS" c5 (V) the c "ndi ti on s of lemma 4 are S" is = po(V) = sq tis f ied. Eence o(V) is a finite module over S'' and the c-inditions of theorem 13 are fulfilled implying that B is a homomorphic 33 REFERENCI:S 1. .J.:.:.H. Datho, Noncommutative Semi-local and Local 2. ~.H. 3. I. s. Cohen, On the Structure 4. C.W. Curtis, The Structure of Nonsemisimnle Duke 1'13. th. tT. 21 ( 195h )-79-85 5. L. Fuchs, Abellan Groups, 6. A.W. Goldie, Localization in Nonc~mmutative Noetherian Rin5s, J. Algebra 5(1967) 89-105 7. G. Hochschild, On the Structure of Algebras with Nonzero Radical, Dull. A111er. '.'<ath. Soc. 53 Tl947) 369-377 ,.., Duke Kath. J. Hin~s, 24(1957) 163-172 Batho, A Note on a Theorem of I.S. Cohen, Portugaliae x~th. 18(1959) 187-192 Ideal Theory of ComTrans. Amer. ~~a th. ~nd plete Local Rine:;s, Soc. 59 ( 19%) 5L~-l06 Alg~br§!:E!_, Budapest, 1958 o. G. Hochschild, Double Vector Spaces over Di_"l{_i_s1_C2_n_ Rings, .t.\rner. J. ?.~ath. 71(1949) 443-460 9. G. Hochschild, Note on the Maximal Algebra, Amer. Math. Soc. 1(1950) 11-1!+ Proc. 10. N. Jacobs,·m, An .Sxten~lon_of G~J.ois 'l'l}_£.2£Y_~o i\Tonnorma.l and Non-separable Fields, A.mer. ,J. Math. 6i'':(i94h) 1.:29 ---