ON UNIVERSAL LOCALIZATION AT SEMIPRIME GOLDIE IDEALS

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1 ON UNIVERSAL LOCALIZATION AT SEMIPRIME GOLDIE IDEALS JOHN A. BEACHY Deparmen of Mahemaical Sciences Norhern Illinois Universiy DeKalb IL 6115 U.S.A. Absrac In his paper we consider an alernaive o Ore localizaion a a semiprime ideal S of a lef Noeherian ring R. In [5 P.M.Cohn inroduced he universal Σ(S)-invering ring for he se Σ(S) of all square marices over R ha remain regular on reducion modulo S. We give an accoun of his universal localizaion from an approach ha uses general ring heoreic echniques raher han hose of he heory of free rings. Togeher wih a review of a number of known resuls we presen a simplificaion of Malcolmson s consrucion ([8) of he universal Σ-invering ring ha makes use of properies paricular o his siuaion. We firs recall some known resuls when P is a prime ideal of a commuaive Noeherian ring R. We will use J(R) o denoe he Jacobson radical of a ring R. Le R P denoe he localizaion of R wih respec o he muliplicaive se R \ P wih canonical homomorphism λ : R R P. I is well-known ha R P /J(R P ) is isomorphic o he field of quoiens of R/P. Furhermore if φ : R T is any ring homomorphism such ha P = φ 1 (J(T )) and he induced mapping φ : R/P T/J(T ) is he embedding of R/P in is field of fracions hen φ(c) + J(T ) is inverible in T/J(T ) for all elemens c R \ P. Since an elemen inverible modulo he Jacobson radical is inverible i follows ha φ(c) is inverible in T for all c R \ P. By he definiion of R P here is a unique homomorphism θ : R P T wih θλ = φ. I is his propery ha we will use o define he universal localizaion of a noncommuaive ring R a a semiprime ideal S. 1 Properies of he universal localizaion Throughou his secion R will denoe a lef Noeherian ring (wih ideniy) and S will denoe a semiprime ideal of R. Consider he following condiions for a ring T and ring homomorphism φ : R T. J 1 : The ring T/J(T ) is a semisimple Arinian ring. J 2 : S = φ 1 (J(T )) 1

2 J 3 : The ring T/J(T ) is a classical ring of lef quoiens of R/S under he induced embedding φ : R/S T/J(T ). J 4 : If θ : R T is a ring homomorphism such ha condiions J 1 J 2 and J 3 are saisfied hen here exiss a unique ring homomorphism θ : T T such ha θ = θ φ. Since condiion J 4 saes ha T is universal wih respec o condiions J 1 hrough J 3 a sandard argumen shows ha if here exiss a ring saisfying condiions J 1 hrough J 4 hen i mus be unique. Before considering he exisence of such a ring we give he relevan definiion. Definiion 1.1 Le R be a lef Noeherian ring wih semiprime ideal S. A ring saisfying he above condiions J 1 hrough J 4 is called he universal localizaion of R a S and will be denoed by R S wih canonical homomorphism λ : R R S. For any ideal I of R he se of elemens c R ha are regular modulo I will be denoed by C(I). We need o exend his definiion relaive o S as follows. For any posiive ineger n le Σ n (S) denoe he se of all marices C such ha C belongs o he n n marix ring M n (R) and he image of C in M n (R/S) is a regular elemen. This will be abbreviaed by saying ha C is regular modulo S. Noe ha C Σ n (S) if and only if he image of C is inverible under he canonical mapping from M n (R) ino he lef classical quoien ring Q cl (M n (R/S)) = M n (Q cl (R/S)). The union over all n > of Σ n (S) will be denoed by Σ(S). The universal localizaion R Σ(S) of R a Σ(S) is defined as he universal Σ(S)- invering ring. I can be consruced as follows (see [4 and [5 for deails). For each n and each n n marix [c ij in Σ(S) ake a se of n 2 symbols [d ij and ake a ring presenaion of R Σ(S) consising of all of he elemens of R as well as all of he elemens d ij as generaors; as defining relaions ake all of he relaions holding in R ogeher wih all of he relaions [c ij [d ij = I and [d ij [c ij = I which define all of he inverses of he marices in Σ(S). Theorem 1.2 Le R be a lef Noeherian ring. For any semiprime ideal S of R he universal localizaion R S exiss and is unique up o isomorphism. Proof. The uniqueness follows immediaely from he definiion. If Σ(S) is he se of all square marices ha are regular modulo S hen Theorem 4.1 of [4 shows ha he universal Σ(S)-invering ring R Σ(S) saisfies properies J 1 hrough J 3. If φ : R T is any ring ha saisfies condiions J 1 hrough J 3 hen for any marix C Σ n (S) i follows ha φ(c) is inverible modulo M n (J(T )) = J(M n (T )) and hence φ(c) is inverible in M n (T ). Since R Σ(S) is he universal Σ(S)-invering ring i saisfies condiion J 4. Proposiion 1.3 Le R be a lef Noeherian ring. If S is a localizable semiprime ideal of R hen he universal localizaion R S coincides wih he Ore localizaion of R a S. 2

3 Proof. If C(S) saisfies he lef Ore condiion i is well-known ha he ring of lef quoiens of R wih respec o he muliplicaive se C(S) saisfies condiions J 1 hrough J 3. Since his ring of lef quoiens is universal wih respec o invering elemens in C(S) he argumen used in he proof of he previous heorem can be repeaed. Proposiion 1.4 Le R be a lef Noeherian ring wih semiprime ideal S. (a) The canonical mapping λ : R R S is an epimorphism in he caegory of rings. (b) The ring R S is fla as a righ module over R if and only if S is a lef localizable ideal. Proof. Par (a) follows from he characerizaion of R S as he universal Σ(S)- invering ring. Par (b) is Corollary 3.2 of [1. Theorem 1.5 Le R be lef Noeherian le N be he prime radical of R and le K = ker(λ) for he canonical homomorphism λ : R R S. (a) The kernel K is he inersecion of all ideals I N such ha C(N) C(I). (b) The ring R/K is a lef order in a lef Arinian ring and R N is naurally isomorphic o Q cl (R/K). Proof. Pars (a) and (b) are Proposiion 1.3 and Theorem 1.4 of [2 respecively. I is shown in Example 4 of [1 ha he universal localizaion a a semiprime ideal of a lef Noeherian ring need no be lef Noeherian. In fac he ring given as an example is a Noeherian ring finiely generaed (as a module) over is cener. On he oher hand i is possible o deermine condiions under which he universal localizaion is lef Arinian. Corollary 1.6 Le R be lef Noeherian le S be a semiprime ideal of R and le K = ker(λ) for he canonical homomorphism λ : R R S. (a) The universal localizaion R S is lef Arinian if and only if S n K for some n >. (b) If P is a minimal prime ideal of R hen R P is lef Arinian. Proof. See Theorem 1.5 and Corollary 1.6 of [1. The symbolic powers of S will be defined as in he commuaive siuaion by exending S n o R S λ(s n )R S and hen conracing back o R. Definiion 1.7 Le R be a lef Noeherian ring wih semiprime ideal S and le λ : R R S be he canonical homomorphism. The n h symbolic power of S denoed by S (n) is defined as S (n) = λ 1 (R S λ(s n )R S ). 3

4 Proposiion 1.8 Le R be a lef Noeherian ring wih semiprime ideal S and le λ : R R S be he canonical homomorphism. (a) S (n) = λ 1 (J(R S ) n ). (b) S (n) is he inersecion of all ideals I such ha S n I S and C(S) C(I). (c) C(S) is a lef Ore se modulo S (n). Proof. Pars (a) and (b) follow from Proposiion 2.2 of [2. Since S/S (n) is he prime radical of R/S (n) par (c) follows from par (b) and Small s Theorem. A number of addiional resuls can be proved for he symbolic powers of S. For example for all posiive inegers n m we have S (n) S (m) S (n+m). For commuaive Noeherian rings i is a sandard resul ha ker(λ) = n=1p (n). This fails in he noncommuaive seing as shown by he following example. Le R be he ring of lower riangular 2 2 marices wih enries from he raional numbers in which he firs enry on he diagonal has odd denominaor. If S is he Jacobson radical of R hen R/S is semisimple Arinian and so R S = R and S (n) = S n for all n. Thus ker(λ) = () n=1s (n). The following proposiion gives some posiive informaion along hese lines. Proposiion 1.9 Le R be a lef Noeherian ring wih semiprime ideal S and le λ : R R S be he canonical homomorphism. Then ker(λ) n=1s (n). Proof. This follows from he fac ha he symbolic power S (n) is he kernel of he canonical homomorphism from R ino R S /J(R S ) n and his homomorphism saisfies properies J 1 hrough J 3 in he definiion of R S. Given a prime ideal P of a wo-sided Noeherian ring R and any posiive ineger n he lef symbolic powers H n of P are defined by Goldie [6 as follows: H 1 = P and by inducion H n is defined as he wo-sided C(P ) closure of P H n 1. Lemma 2.3 of [2 shows ha P (n) = H n for any posiive ineger n. Assume ha R is Noeherian and le P be a prime ideal of R. For each posiive ineger n le Q n be he Arinian classical ring of quoiens of R/P (n). Then here is a canonical epimorphism Q n+1 Q n for n = Le Q be he inverse limi of he rings {Q n } n=1 under hese epimorphisms and le µ : R Q be he induced homomorphism. Goldie s localizaion Q of R a P is defined as he inersecion of all subrings Q of Q such ha Q /J(Q ) is simple Arinian J(Q ) n = () and µ(p ) J(Q ) J( Q). The proof of Theorem 1 of [7 shows ha Q/J(Q) = Q cl (R/P ) n=1j(q) n = () and P (n) = µ 1 (J(Q) n ). Theorem 1.1 Le P be a prime ideal of he Noeherian ring R. localizaion of R a P is isomorphic o R P / n=1 J(R P ) n. Then Goldie s Proof. See Theorem 2.4 of [1. 4

5 2 Equivalence of quoiens Throughou his secion R X will denoe a fixed lef R-module and he direc sum of n copies of X will be denoed by X n. The noaion x X n will be used o denoe a row vecor wih enries in X and he corresponding column vecor will be denoed by. The ideniy of M n (R) will be denoed by I n ; he subscrip will be omied when he size is clear from he conex. Throughou he remainder of he paper Σ will denoe a se of square marices over R such ha (i) Σ conains all permuaion marices; [ C A (ii) if C D Σ hen Σ for any marix A of he appropriae size; and D (iii) if C D Σ and CD is defined hen CD Σ. We noe ha if R is lef Noeherian and S is a semiprime ideal of R hen he se Σ(S) of all square marices regular modulo S saisfies he above condiions. For he given se Σ an elemen x X is said o be Σ-orsion if x is an enry of a column vecor v wih enries in X such ha Cv = for some C Σ. The proof of Proposiion 2.1 of [5 shows ha he se of all Σ-orsion elemens of X is a submodule which we denoe by rad Σ (X). Then X is said o be Σ-orsion if rad Σ (X) = X. I should be noed ha if R is lef Noeherian S is a semiprime ideal of R and X is finiely generaed hen i is possible o give anoher characerizaion of Σ(S)- orsion modules. By Proposiion 1.1 of [2 X is Σ(S)-orsion if and only if X/SX is a orsion module over R/S. The elemens of a module of quoiens denoed by X Σ will be consruced as equivalence classes of ordered riples (a C ) where a R n C Σ n and x X n (for any posiive n). The ordered riples are modeled on he elemen ac 1 where C is inverible as would be he case over R Σ. Le a R n C Σ n x X n b R m D Σ m y X m and assume ha U V are inverible n n marices. If C and D are inverible hen we have he following ideniies. (i) [a b [ C D 1 [ x y (ii) au(v CU) 1 V = ac 1 (iii) ac 1 = = C 1 = ac 1 + bd 1 y An addiion of riples is based on he firs of hese ideniies. The second moivaes he definiion of he iniial equivalence relaion for riples. We say ha (a C ) (b D y ) if here exis inverible marices U V in Σ such ha b = au D = V CU and y = V. I is easily checked ha his defines an equivalence relaion. (The proof of ransiiviy uses he fac ha Σ is closed under producs.) We noe ha (a C ) (b D y ) only if C and D have he same size. Equaion (iii) provides he moivaion for he definiion of he subsemigroup ha induces he final equivalence relaion. 5

6 Definiion 2.1 Le (a C ) (b D y ) be ordered riples wih a b R n C D Σ n and x y X n for some posiive ineger n. If here exis inverible n n marices U V in Σ such ha b = au D = V CU and y = V hen we say ha (a C ) is congruen o (b D y ) via U V wrien (a C ) (b D y ) via U V. For a R n C Σ n and x X n he noaion (a : C : ) will be used for he equivalence class of he ordered riple (a C ) under he equivalence relaion. The se of all such equivalence classes for all posiive inegers n will be denoed by Σ 1 X. The subse of Σ 1 X consising of all equivalence classes of elemens of he form (e 1 E 1 ) or ( E 2 e 2) or ( [ E1 [e 1 E 2 [ for some e 1 R m E 1 Σ m E 2 Σ n and e 2 X n will be denoed by Σ 1 X. e 2 ) Proposiion 2.2 The sum of elemens (a : C : ) (b : D : y ) Σ 1 X defined by ( [ [ ) C x (a : C : ) + (b : D : y ) = [a b : : D y yields an associaive commuaive binary operaion on Σ 1 X. Proof. If (a 1 C 1 1) (a 2 C 2 2) via inverible marices U V Σ and (b 1 D 1 y1) (b 2 D 2 y2) via inverible marices U V [ Σ hen i is easy [ o check U V ha he respecive sums are congruen via he marices U and V which are inverible and belong o Σ. Thus addiion of equivalence classes is welldefined on Σ 1 X and i is associaive by definiion. Using permuaion marices i is sraighforward o check ha addiion of equivalence classes is commuaive. Proposiion 2.3 For elemens x y Σ 1 X he relaion defined by x y if here exis z 1 z 2 Σ 1 X such ha x + z 1 = y + z 2 is a congruence on he semigroup Σ 1 X. The se Σ 1 X/ of equivalence classes of his congruence is an abelian group. Proof. Using permuaion marices i is easy o check ha Σ 1 X is closed under addiion. Since addiion in Σ 1 X is associaive and commuaive i follows easily ha is a congruence. Therefore addiion of equivalence classes in Σ 1 X/ is welldefined and saisfies he associaive and commuaive laws. The equivalence class of Σ 1 X is he zero elemen and he following compuaion shows he exisence of addiive inverses. 6

7 For any elemen (a : C : ) Σ 1 X we have ( [ C (a C ) + (a C ) = [a a C Since [a a [ [ [ I I I I C = [a [ [ I [ I C I I x and I = we have via [ I I I (a C ) + (a C ) [ I I I ( [ C [a C [ [ I I I ) = [ and he las elemen belongs o Σ 1 X.. [ C C If C 1 C 2 are inverible marices such ha C 2 A 1 = A 2 C 1 for marices A 1 A 2 hen A 1 C1 1 = C2 1 A 2 and so aa 1 C1 1 = ac2 1 A 2. This moivaes he following lemma for riples (aa 1 : C 1 : ) and (a : C 2 : A 2 ) such ha C 2 A 1 = A 2 C 1 a siuaion reminiscen of he lef Ore condiion. This lemma will prove o be very useful compuaionally. Lemma 2.4 Le a R m C 1 Σ n x X n and le A 1 be any m n marix over R. If here exis an m n marix A 2 and a marix C 2 Σ m such ha C 2 A 1 = A 2 C 1 hen (aa 1 : C 1 : ) (a : C 2 : A 2 ). Proof. and If a C 1 C 2 A 1 A 2 and x are as saed hen [ [ [a aa 1 Im A 1 Im A = [a 2 I n I n [ Im A 2 I n Therefore ( [ C2 [a aa 1 C 1 We hen have [ C2 C 1 [ [ Im A 1 ) I n = [ [ C2 ( [ C2 [a C 1 ) [ A2 x = C 1. [ A2 x (aa 1 : C 1 : ) (a : C 2 : ) + (aa 1 : C 1 : ) ( [ [ ) C2 = [a aa 1 : : C 1 ( [ [ ) C2 A2 x = [a : : C 1 = (a : C 2 : A 2 ) + ( : C 1 : ) (a : C 2 : A 2 ). 7 ).

8 This complees he proof. If S is a semiprime ideal of R for which C(S) is a lef denominaor se hen for each (a : C : ) Σ(S) 1 X here exis elemens y X and d C(S) such ha (a : C : ) (1 : d : y). To see his le λ : R R S be he classical lef localizaion of R a C(S). We can assume wihou loss of generaliy ha λ is one-o-one. Le (a : C : ) Σ(S) 1 X. Then λ(c) is inverible over R S so i is possible o find a common denominaor d C(S) for he enries of λ(a)λ(c) 1. Thus we have elemens d C(S) and b R n such ha da = bc and hen i follows from Lemma 2.4 ha (1 a : C : ) (1 : d : b ). Proposiion 2.5 Le a R n C Σ n and x X n. (a) If b R n and y X n hen and (a : C : ) + (a : C : y ) (a : C : (x + y) ) (a : C : ) + (b : C : ) (a + b : C : ). (b) For any marices P Q such ha P C CQ Σ n (a : C : ) (a : P C : P ) and (a : C : ) (aq : CQ : ). (c) For any b R m D Σ m y X m and any marices A B of he appropriae size ( [ [ ) C A x (a : C : ) [a b : : D and ( [ [ ) D B y (a : C : ) [ a : : C. Proof. (a) Since C [I I = [I I [ C C ( [ [ ) C x a [I I : : C y i follows from Lemma 2.4 ha ( a : C : [I I and so (a : C : ) + (a : C : y ) (a : C : (x + y) ). The second half of condiion (a) follows in a similar fashion. (b) By Lemma 2.4 we have (ai : C : ) (a : P C : P ) since (P C)(I) = (P )(C). Similarly (C)(Q) = (I)(CQ) shows ha (aq : CQ : ) (a : C : I ). (c) Since [ C A D ( [a b [ I [ I = [ I ) : C : C by Lemma 2.4 we have ( [ C A [a b : D 8 : [ x [ I y ) ).

9 Finally since C [ I = [ I [ D B C ( [ D B a [ I : C [ ) y : i again follows from Lemma 2.4 ha ( a : C : [ I [ y ). This complees he proof. Proposiion 2.6 Le (a : C : ) (b : D : y ) Σ 1 X. Then (a : C : ) (b : D : y ) if and only if here[ exis vecors e 1 u over R and e 2 v over X (of he E1 appropriae size) and marices and P Q Σ such ha uv E = and 2 ( [ [ ) (a C ) + (b D y E1 ) + [e 1 E 2 e = (uq P Q P v ). 2 Proof. Firs assume ha (a : C : ) (b : D : y ). Then (a : C : ) + (b : D : y ) is equivalen o zero since Σ 1 X/ is an abelian group and (b : D : y ) represens he addiive inverse of (b : D : y ). By definiion of ( here exis [ z 1 z 2 Σ 1 [ X such ) ha (a : C : ) + (b : D : y F1 ) + z 1 = z 2. If z 2 = [f 1 : : F 2 f2 hen using he definiion of he equivalence relaion here exis inverible marices U V Σ of he appropriae size such ha ( [ [ ) (a C ) + (b D y F1 ) + z 1 = [f 1 U V U V. F 2 Since[ z 1 already has he desired form [ we only need o facor he righ hand side. Le I F1 Q = U and P = V. This facorizaion yields (uq P Q P v F 2 I ) for u = [f 1 and v = [ f 2 and hen uv =. Conversely suppose ha he given condiion holds. Then by he definiion of we have (a : C : )+(b : D : y ) (uq : P Q : P v ). Using he previous proposiion and Lemma 2.4 we have (uq : P Q : P v ) (u : I : v ) (1 : 1 : uv ) = (1 : 1 : ) f 2 and hus (a : C : ) (b : D : y ). Recall ha an elemen x X is Σ-orsion if i is an enry in a vecor v X n such ha Cv = for some C Σ. Since Σ is closed under producs (when defined) and conains all permuaion marices i can be assumed ha x is he firs enry of v. 9

10 Theorem 2.7 Le x y X. (a) In Σ 1 X (1 : 1 : x) (1 : 1 : y) if and only if x = av and y = bw for some a b R n v w X n and some n > such ha here exis C D P Q Σ n saisfying ad = bq Cv = P w and CD = P Q. (b) Furhermore x y rad Σ (X) if and only if in condiion (a) i is possible o ake a = b C = P and D = Q = I. Proof. (a) If (1 : 1 : x) (1 : 1 (: y) hen [ here exis [ z 1 z) 2 Σ 1 X such ha E1 (1 1 x) + z 1 (1 1 y) + z 2. If z 1 = [e 1 E 2 e hen (1 1 x) + z 1 2 has he form 1 x [1 e 1 E 1. E 2 e 2 This can be facored in he form (ad CD Cv ) for 1 1 a = [1 e 1 C = E 1 D = I I E 2 v = [x e 2 wih av = x. A similar facorizaion of (1 1 y) + z 2 muliplied by he inverible marices obained from he definiion of he relaion gives (ad CD Cv ) = (bq P Q P w ) where bw = y. Conversely if he saed condiion holds hen we have (1 : 1 : x) (a : I : v ) (ad : CD : Cv ) = (bq : P Q : P w ) (b : I : w ) (1 : 1 : y). (b) If x y X wih x y rad Σ (X) hen x y is he firs enry of a vecor u such ha Cu = for some C Σ. I is possible o wrie u = v w where x and y are he firs enries of v and w respecively so ha Cv = Cw. If a denoes he vecor over R wih 1 as is firs enry and zeroes elsewhere hen av = x and aw = y giving he desired resul. Conversely if he condiion is saisfied hen C(v w ) = so all enries of v w belong o rad Σ (X). I follows ha x y = a(v w ) rad Σ (X) compleing he proof. Malcolmson [8 has shown ha λ : R R Σ given by λ(r) = (1 : 1 : r) is a ring homomorphism which invers he marices in Σ. I follows immediaely ha any Σ-orsion elemen (orsion on eiher lef or righ) mus be mapped o zero by λ. The following example shows since he lef Σ-orsion ideal differs from he righ Σ-orsion ideal ha i is possible o have equivalen elemens (1 : 1 : r) and (1 : 1 : s) for which r s does no belong o he lef Σ-orsion ideal. 1

11 Le R be he following ring of lower riangular marices wih enries from he ring of inegers or he ring of inegers modulo 2 as indicaed. Le S be he prime radical of R le Σ be he se Σ(S) and consider he ideal I defined below. R = Z 2 Z 2 Z Z 2 Z 2 Z 2 S = Z 2 Z 2 Z 2 I = Z 2 Z 2 A marix belongs o C(S) if and only if is enries on he main diagonal are all 1 nonzero so 2 C(S). This elemen annihilaes I/SI which shows by 1 Proposiion 1.1 of [2 ha I is lef Σ(S)-orsion. Furhermore I is he lef Σ(S)-orsion ideal since S/S 2 is no Σ(S)-orsion. Arguing by symmery he righ Σ(S)-orsion ideal is he boom row of S so he kernel of λ mus be S. Then (1 : 1 : x) is equivalen o (1 : 1 : ) for he elemen x = bu x is no Σ(S)-orsion. This esablishes a clear disincion 1 beween condiions (a) and (b) of Theorem Modules of quoiens Throughou his secion R X will denoe a fixed lef R-module. We begin wih he definiion of a module of quoiens. Definiion 3.1 The se of equivalence classes of Σ 1 X/ will be denoed by X Σ. The noaion [a : C : will be used for he class of (a : C : ) Σ 1 X. Proposiion 2.6 of Secion 2 shows ha our definiion of equivalence for elemens of R Σ is he same as ha of Malcolmson [8. The muliplicaion abou o be defined coincides wih ha in Malcolmson s consrucion so we have in fac defined he universal Σ-invering ring. Thus properies of R Σ may be used in consrucing modules of quoiens. I should be noed ha he scalar muliplicaion defined below can be used o consruc he ring R Σ and he necessary properies can be verified using only he echniques of his paper. Le a r R n C Σ n b R m D Σ m and y X m. If C D are inverible hen we have he following ideniy which moivaes he definiion of a scalar muliplicaion. [a [ C r b D 1 [ y = ac 1 r bd 1 y 11

12 Proposiion 3.2 The scalar produc of elemens (a : C : r ) Σ 1 R and (b : D : y ) Σ 1 X defined by ( [ [ ) C r (a : C : r ) (b : D : y ) = [a : b : D y yields a well-defined associaive operaion. Proof. If (a 1 C 1 r1) (a 2 C 2 r2) via inverible marices U V Σ and (b 1 D 1 y1) (b 2 D 2 y2) via inverible marices U V Σ hen we have ( [ [ ) ( [ [ ) C1 r [a 1 1b 1 C2 r D 1 y1 [a 2 2b 2 D 2 y2 [ [ U V via he marices U and V. If p q Σ 1 R and x Σ 1 X hen p(q x) = (p q)x as a consequence of he way in which marices are combined. Lemma 3.3 The following condiions hold for scalar muliplicaion. (a) If (a : C : r ) Σ 1 R and (1 : 1 : x) Σ 1 X hen (a : C : r )(1 : 1 : x) (a : C : r x). (b) If (1 : 1 : s) Σ 1 R and (a : C : ) Σ 1 X hen (1 : 1 : s)(a : C : ) (sa : C : ). Proof. (a) Since C [I = [I [ r C r we have 1 ( [ [ ) C r (a : C : r )(1 : 1 : x) = a [I : : 1 x ( [ ) [I r a : C : = (a : C : r x). x [ (b) Since [ 1 sa C [ sa I = C we have I ( [ ) [1 sa (sa : C : ) = : C : I ( [ [ 1 sa [1 : : C I = (1 : 1 : s)(a : C : ) ) This complees he proof. 12

13 Lemma 3.4 Le p = (u : P : r ) q = (v : Q : s ) Σ 1 R and le x = (a : C : ) y = (b : D : y ) Σ 1 X. Then Proof. (p + q)x p x + q x and q(x + y) q x + q y. We have he following equaliies. [u v (p + q)x = p x + q x = [u v : I I I I : P r a C Q s a C P r a Q s a C : The wo expressions are equal by Lemma 2.4 since P r a I I C I Q s a I = I I C I I I I I I : P r a Q s a C Finally by Lemma 2.4 he expressions given below for q(x + y) and q x + q y are equal since we have he following ideniy. Q s a s Q s b I I I I a C I = I C Q s D I I b D q x + q y = [v q(x + y) = [v : This complees he proof. I I I I : Q s a s b C D Q s a C Q s b D : I I I I : y. y Theorem 3.5 For any module R X he se X Σ is a unial lef module over R Σ. 13

14 Proof. We have shown in Proposiion 2.3 ha X Σ is an abelian group under addiion. To show ha he muliplicaion defined in Proposiion 3.2 respecs he equivalence relaion on Σ 1 X le p q Σ 1 R and x y Σ 1 X wih p q and x y. Then here exis z 1 z 2 Σ 1 R and z 3 z 4 Σ 1 X wih p + z 1 = q + z 2 and x + z 3 = y + z 4. Thus we have (p + z 1 )x = (q + z 2 )x and q(x + z 3 ) = q(y + z 4 ). I follows from Lemma 3.4 ha p x + z 1 x q x + z 2 x and q x + q z 3 q y + q z 4. By definiion of scalar muliplicaion we have z 1 x z 2 x q z 3 q z 4 Σ 1 X and so we obain p x q y. The disribuive laws hold by Lemma 3.4. Finally X Σ is a unial lef R Σ -module by Lemma 3.3. Proposiion 3.6 For he module R X define he mapping η : X X Σ by η(x) = [1 : 1 : x for all x X. Then η is an R-homomorphism. Proof. The ring R acs on X Σ via he homomorphism λ : R R Σ defined by λ(r) = [1 : 1 : r. Thus by Lemma 3.3 (a) for any r R and any x X we have η(rx) = [1 : 1 : rx = [1 : 1 : r[1 : 1 : x = λ(r)η(x). Proposiion 3.7 Le [a : C : X Σ. Then [a : C : = λ(a)λ(c) 1 η( ) for he canonical mappings λ : R R Σ and η : X X Σ. Proof. Le e i denoe he vecor wih 1 in he ih enry and zero elsewhere. Assume ha C Σ n and le C = [c ij for elemens c ij R. I follows from Lemma 3.3 (b) ha for a fixed k > [1 : 1 : c ki [e i : C : e j = i=1 [c ki e i : C : e j i=1 = [e k C : C : e j = [e k : I : e j = [1 : 1 : e k e j = δ kj. (Proposiion 2.5 (a) and (b) and Lemma 2.4 have also been used.) A similar argumen holds on he oher side showing ha he enries of λ(c) 1 are jus he elemens [e i : C : e j in R Σ. Having found λ(c) 1 i follows ha λ(a)λ(c) 1 η( ) = ( ) [1 : 1 : a i [e i : C : e j [1 : 1 : x j j=1 i=1 14

15 = = ( ) [a i e i : C : e j [1 : 1 : x j j=1 i=1 [a : C : e j[1 : 1 : x j = j=1 j=1 = [a : C :. [a : C : e jx j This complees he proof. We say ha he module R X is Σ-orsionfree if C = implies x = for all C Σ n and all x X n. We say ha X is Σ-divisible if for each x X n and each C Σ here exiss y X n such ha Cy =. Theorem 3.8 The homomorphism η : X X Σ is an isomorphism if and only if X is Σ-orsionfree and Σ-divisible. Proof. If η is an isomorphism hen X has a naural srucure as a lef R Σ -module and so C = implies = λ(c) 1 C = for any x X n showing ha X is Σ-orsionfree. Similarly X is Σ-divisible since for any x X n we have = C(λ(C) 1 ). Conversely suppose ha X is Σ-orsionfree and Σ-divisible. For any elemen [a : C : X Σ here exiss y such ha = Cy and hen [a : C : = [a : C : Cy = [a : I : y = [1 : 1 : ay by Proposiion 2.5 (b) and Lemma 2.4. Thus [a : C : = η(ay ) and so η is an epimorphism. Now le x ker(η). By Theorem 2.7 (a) here exis a b R n v w X n and C D P Q Σ n such ha x = av bw = ad = bq Cv = P w and CD = P Q. Since X is Σ-divisible here exis v 1 w 1 X n such ha v = Dv1 and w = Qw1. Therefore CDv1 = Cv = P w = P Qw1 and so v1 = w1 since CD = P Q and X is Σ-orsionfree. Bu hen x = av = a(dv1) = (bq)w1 = bw = and so η is a monomorphism. As in Theorem 4.9 of [9 he following corollary implies ha if R is a lef herediary ring hen so is he universal localizaion R Σ. The corollary can be proved using an argumen similar o he sandard one for he class of orsionfree divisible modules over an inegral domain. Corollary 3.9 In he caegory of lef R-modules he class of lef R Σ -modules is closed under exensions. Theorem 3.1 For any module R X he module of quoiens X Σ is naurally isomorphic o R Σ R X. 15

16 Proof. Le ɛ : X R Σ R X be he naural homomorphism defined by ɛ(x) = 1 x for all x X. Then since X Σ is an R Σ -module we can define θ : R Σ R X X Σ wih θ(q x) = qη(x) for all q R Σ and x X where η is he canonical mapping from X o X Σ. Then θɛ = η and for x X and λ(a)λ(c) 1 λ(r ) R Σ we have θ ( λ(a)λ(c) 1 λ(r ) x ) = λ(a)λ(c) 1 λ(r )η(x) i=1 = λ(a)λ(c) 1 η(r x). I follows ha given λ(a)λ(c) 1 η( ) X Σ we have ( ) λ(a)λ(c) 1 η( ) = θ λ(a)λ(c) 1 λ(e i) e i which shows ha θ is ono. To show ha θ is one-o-one we will show ha i has an inverse. For (a : C : ) Σ 1 X define φ((a : C : )) = λ(a)λ(c) 1 λ(e i) e i. i=1 I is clear ha φ is well-defined on Σ 1 X and addiive. To show ha i is well-defined on X Σ by Proposiion 2.6 i suffices o show ha φ((a : C : )) = for any elemen (a : C : ) of he form (uq : P Q : P v ) wih P Q Σ and uv =. We have φ((uq : P Q : P v )) = = = = λ(uq)λ(p Q) 1 λ(e i) e i P v i=1 ( ) λ(uq)λ(p Q) 1 λ(e i) (e i P e j)e j v i=1 j=1 λ(u)λ(q)λ(q) 1 λ(p ) 1 λ(p )λ(e j) e j v j=1 λ(u)e j e j v = j=1 j=1 1 ue je j v = 1 uv. This expression is equal o zero whenever uv =. I can be shown easily ha φθ = 1 and so θ is an R Σ -isomorphism. I is clear ha θ is a naural ransformaion. References [1 J. A. Beachy Inversive localizaion a semiprime Goldie ideals Manuscripa Mah. 34 (1981)

17 [2 J. A. Beachy On noncommuaive localizaion J. Algebra 87 (1984) [3 J. A. Beachy and W. D. Blair Examples in noncommuaive localizaion J. Algebra 99 (1986) [4 P. M. Cohn Free rings and heir relaions (2nd ed. Academic Press London New York 1985). [5 P. M. Cohn Inversive localisaion in Noeherian rings Commun. Pure Appl. Mah. 26 (1973) [6 A. W. Goldie Localizaion in non-commuaive Noeherian rings J. Algebra 5 (1967) [7 A. W. Goldie A noe on noncommuaive localizaion J. Algebra 8 (1968) [8 P. Malcolmson Consrucion of universal marix localizaions Lecure Noes in Mah V 951 (Springer Verlag Berlin Heidelberg New York 1982) pp [9 A. H. Schofield Represenaions of rings over skew fields London Mahemaical Sociey Lecure Noes 92 (Cambridge Universiy Press Cambridge 1985). [1 B. Sensröm B. Rings of Quoiens (Springer Verlag Berlin Heidelberg New York 1975). 17