ON FREE RING EXTENSIONS OF DEGREE N

Transcription

1 I terat. J. Math. & Mah. Sci. Vl. 4 N. 4 (1981) ON FREE RING EXTENSIONS OF DEGREE N GEORGE SZETO Mathematics Departmet Bradley Uiversity Peria, Illiis U.S.A. (Received Jue 25, 1980) ABSTRACT. Nagahara ad Kishimt [i] studied free rig extesis B(x) f degree fr sme iteger ver a rig B with i, where x b, cx xp(c) fr all c ad -I sme b i B (p autmrphism f B), ad {i, x x is a basis. Parimala ad Sridhara [2], ad the authr ivestigated a class f free rig extesis called geeralized quaterl algebras i which b -I ad p is f rder 2. The purpse f the preset paper is t geeralize a characterizati f a geeralized quateri algebra t a free rig extesi f degree i terms f the Azumaya algebra. Als, it is shw that a e-t-e crrespdece betwee the set f ivariat ideals f B uder p ad the set f ideals f B(x) leads t a relati f the Galis extesi B ver a ivariat subrig uder t the ceter f B. KEY WODS AND PHRASES. Free rig extesis, separable algebras, Azumaya algebras, Gais extesis. 198u MATHEMATICS SUBJECT CLASSIFICATION CODES: 16A16, 13A20, 13B05. I. INTRODUCTION. Kishimt [3], ad Nagahara ad Kishimt [i] studied free rig extesis f

2 704 G. SZETO degree 2 ad fr a iteger > 2: (I) B(x) is a free rig extesi ver a 2 rig B with i with a basis {i, x} such that x xa + b fr sme a ad b i B, ad cx x0(c) fr each c i B, where p is a rig autmrphism f B f rder 2. (2) B(x), a free rig extesi f degree > 2 is similarly defied with a basis {i, x, x }, ad x b fr sme b i B ad cx xp(c) fr each c i B, where 0 is f rder. Sme special free rig extesis called geeralized quateri algebras were ivestigated by Parimala ad Sridhara [2] ad the authr Szet ([4] [5]). Oe f their results is a characterizati f the Galis extesi f B ver a subrig ([2], Prpsiti 1.1): Let B(x) be a geeralized quateri algebra (x 2 -I) ver a cummutative rig B with 2 a uit i B. The B is Galis ver A (={a i B/0(a) a fr a autmrphism 0 f rder 2}) if ad ly if BAB(X) is a matrix algebra f rder 2. The abve characterizati was geeralized t a free rig extesi f degree, B(x) with x -I ([4], Therems 3.4 ad 3.5}. Te purpse f the preset paper Is t ctiue the abve geeralizati t a free rig extesi. Als, we shall shw that there is a e-t-e crrespdece betwee the set f ivariat ideals f B uder O ad the set f ideals f B(x). This crrespdece will lead t a relati f the Galis extesi B ver the ivariat subrig A uder 0 t the ceter Z f B ver A. 2. PRELIMINARIES. Thrughut, we assume that B is a rig (t ecessarily cummutative) with i, p a autmrphism f B f rder fr sme psitive iteger, A {a i B/0(a) a}, ad B(x) a free rig extesi ver B with a basis {i, x, x-l} such that x b ad ax x0(a) fr sme b ad all a i B (hece (b) b ([i], p. 20)). Let T be a rig ctaiig a subrig R with i. The T is called a separable extesi ver R if there exist elemets (ui, v i / i i, m fr sme iteger m} such that a (uivl) (luivi)a fr all a i T is ver R ad.uivi 1 ([6], [7]). Such a elemet luivi is called a separable idemptet fr T. If R is i the ceter f T, the separable extesi T is called a separable R-algebra. I particular, if R is the ceter f T, the separable R-algebra T is called a Azumaya R-a!geb.ra (16], [7]). A cmmutative rig extesi S f R is called a

3 FREE RING EXTENSIONS 705 splittig ri fr the Azumaya R-algebra T if ST HOms(P,P fr a prgeeratr S-mdule P ([6], [7]). The rig extesi T ver R is called a Galis extesi with a fiite autmrphism grup G (Galis grup) if (I) R {a i T / u(a) a fr all u i G}, ad (2) there exist elemets {ui, v i i T / i I,..., m fr sme iteger m} such that (i).uivi i, ad (2).uiu(vi) 0 fr each u the idetity f G ([7], [8]). 3. A GENERAL PARIMALA-SRIDHARAN THEOREM. I this secti, we shall geeralize the Parimala-Sridhara [2] therem t a free rig extesi B(x) f degree fr a iteger such that x b ad ax xp(a) fr sme b ad all a i B where p is a autmrphism f B f rder. We te that if B(x) is separable ver B the b is a uit ([i], Prpsiti 2.4). The cverse hlds if is als a uit: LEMMA 3.1. If ad b are uits, the B(x) is a separable extesi ver B. PROOF. Sice b is i A ([I], p. 20) ad s-ice p the Idetity, it is straightfrward t verify that the elemet u b-i (i=x.ix) satisfies the equatis: au ua fr all a i B(x), ad b -I -I (xix) i, is ver B. We remark here that there are separable extesis with (-- 2) t a uit ([4], Therem 4.2). With the same prf as give fr Prpsiti 1.2 i [7] we have a characterizati fr Galis extesis f -cmmutative rigs: LEMMI 3.2. Let B be a rig extesi f A with a fiite autmrphism grup G such that A B G (={a i B / u(a) a fr each i G}). The B is Galis ver A if ad ly if the left Ideal geerated by {a-u(a) / fr a i B} B fr ay the idetity f G. THEOREM 3.3. Let ad b be uits i B. If B is Galis ver A which is c taied i the ceter, Z f B with a Galis grup {i, 0,.,0 } f rder, the the free rig extesi B(x) f degree is a Azumaya A-algebra, where x b ad cx xp(c) fr each c i B. PROOF. By Lemma 3.1, B(x) is separable ver B. Sice B is Galis ver A, B is separable ver A. Hece B(x) is separable ver A ([4], the prf f Therem 3.4).

4 706 G. SZETO S, lt suffices t shw that the ceter f B(x) is A. Let u a be i the i--o lx ceter. The xu ux. Ntig that {l,x,x is a basis fr B(x) ver B, we have that a i are i A. Als, au ua fr all a i B, s ai(a-pi(a)) 0 fr each i O. Hece the cetral elemets a i are i the left aihilatrs f the left ideal geerated by {a-pi(a) / a B} fr i # O. By hypthesis, B is Galls ver A, s a_ 0 fr each i # 0 by Lemma 3.2. Thus u a i A. Clearly, A is i the ce ter f B(x). Therefre, A the ceter f B(x). By the Parimla-Sridhara therem ([3], Prpsiti i.i), let B(x) be a geeralized quateri algebra (x 2 =-i) ver a cmmutative rig B. The, B is Galis ver A (= {a i B / (a) a fr a autmrphism 0 f rder 2}) if ad ly if BHAB(X) is a matrix algebra f rder 2 ver B. Hece, Therem 3.3 geeralizes the ecessity f the Parlmala-Sridhara therem. Fr the sufficiecy, we first give a e-t-e crrespdece betwee the sets f ideals f B, f B(x), f A, ad the ceter Z f B. A ideal I f B is called a G-ideal if (I) I. Sice -I p(z) Z, a G-ideal J f Z is similarly defied, where G i,0,...,0 }. THEOREM 3.4. Let B(x) be a Azumaya A-algebra. The there exists a e-te crrespdece betwee (i) the set f G-ideals f B, (2) the set f ideals f B(x), ad (3) the set f ideals f A. PROOF. At first, we wat t give a structure f a G-ideal I f B. Sice -i I, xlb(x> c 0 (1)B(x) IB(x). Hece IB(x) is a ideal f B(x). By hypthesls, B(x) is a Azumaya A-algebra, s IB(x) I B(x) where I IB(x) A ([7], Crllary 3.7, p. 54). Ntig that {l,x,...,x } is a basis fr B(x) ver B, we have I I B ad I I A. Next, it is easy t see that J B is a G-ideal f B fr ay ideal J f A. Thus the set f G-ideals f B are i e-t-e crre spdece with the set f ideals f A frm the abve represetati I B f a G- ideal I f B. By hypthesis agai, B(x) is a Azumaya A-algebra, s the set f ideals f B(x) ad the set f ideals f A are i e-t-e crrespdece uder IB(X)++l. fr a ideal I f A. Thus the therem is prved. COROLLARY 3.5. Let ad b be uits i B. Suppse B is Galls ver A which is ctaied i Z. The there exists a e-t-e crrespdece betwee the set f G-ideals f Z ad the set f ideals f B(x).

5 FREE RING EXTENSIONS 707 PROOF. Sice B is Galls ver A, B is a separable A-algebra. Hece B is Azumaya ver its ceter Z ([7], Therem 3.8, p. 55). Thus the set f G-ideals f B ad the set f G-ideals f Z are i e-t-e crrespdece; ad s Therem 3.4 implies the crllary. Nw we shw a geeralizati f the sufficiecy f the Parimala-Srldhara therem. The set {a i B / p(a) a} is deted by Bp. Let G be a autmrphlsm m-i grup, {i,, } btaied frm G (= {i, }) by takig m as the miimal iteger such that 0m the idetity Z. We dete the ideal geerated by {a-i(a) / a i Z} by I i fr i l,...,m-l. It is easy t see that each I i is a G-ideal such that Ira_ 1 c Ira_ 2 c c I I. characterizes the Galls extesi f Z ver A. That is: We shall shw that the chai f li s THEOREM 3.6. If B(x) is a Azumaya A-algebra such that I 1 12 Ira_l, the Z is Galis ver A with a Galls grup G. _ PROOF. I case Z A the therem is trivial. Let Z # A. The m # 0. Clearly, A BG Z G. Nw we assume Z is t Galls ver A. The the ideal I f Z is t Z 1 ([7], Prpsiti 1.2, p. 80) sice I 1 12 Ira_ by 1 hypthesis. Sice I is a 1 G-ideal, I IZ fr sme ideal I f 1 A by Therem 3.4. Hece B(x)/llB(X A/.IAB( is a Azumaya A/l-algegra ([7], Prpsiti i.ii, p. 46). But () i B(x)/llB(X fr each a i Z, s R --. This implies that Z is ctaied i the ceter A/I f the Azumaya A/l-algebra A/IQAB(X). This is impssible sice Z is t ctaied i A. Thus Z is Galis ver A. COROLLARY 3.7. By keepig the tatis f Therem 3.6, if B is Galis ver A with a Galls grup G (= {l,p,..., }) such that I 1 12 Ira_l, the Z is Galis ver A with a Galls grup G, where b ad are uits i B. PROOF. Therem 3.3 implies that B(x) is a Azumaya A-algebra, s the crllary is a csequece f Therem 3.6. As give i Therem 3.6, let B(x) be a Azumaya A-algebra. If B is cmmutative B Z. Nw assume B is t Galis ver A. The there is a I i fr sme i l,...,m-i such that I i # Z. Oe ca shw as give i Therem 3.6 that A/IIQAB(x is a Azumaya algebra such that x i is i the ceter A/I i. Thus we have a ctradfctl. This prves that B is Galls ver A. S, Therem 3.6 geeralizes Therems 3.4 ad

6 708 G. SZETO ad 3.5 i 4]. 4. SPLITTING RINGS. I this secti, we shall shw that if B(x) is a Azumaya A-algebra i which b ad are uits, the A(x) is a splittig rig fr B(x) such that A(x) is a chai f Galis extesis f degree 2 (that is, A(x) A(x2). A(x) A, such that A(xi) is Galis ver A(x21)). THEOREM 4.1. Let A be a cmmutative rig with I, x b i A, ad ax xa fr each a i A. If b ad are uits i A with a pwer f 2 (-2 m fr sme m), the A(x) is a chai f Galls extesis f degree 2. PROOF. We defie a mappig =: A(x)/A(x) by e(x) -x ad a(alxl )" lai(e(x))i fr i 0,i, -l. The it is straightfrward t check that s is a autmrphism f A(x) f rder 2 such that (A(x)) A(x2). Sice m-i 2 (= 2 m 2.2m-l) ad b (= x (x2) 2 are uits i A, 2 ad x are uits i A(x2). Nw we claim that A(x) is Galls ver A(x2) with a Galls grup {I,}. I fact, let a.] (2x2)-ix, a 2 2-1, b I x ad b 2 I. The we have albl+a2b2 i ad al(bl)+a2(b2) 0. Thus A(x) is Galis ver A(x2) f degree 2. Similarly, we ca shw that A(x2) is Galis ver A(x4) with a Galis grup {1,8} with (x =-x f rder 2. Therefre, a iducti argumet ccludes the existece f a chai f Galis extesis f degree 2. Fr the class f free rig extesis B(x) f degree as give i [i], Seci) tl 2 such that c ad (l-c are uits i A where c i ad i 1,2,...,-l, we have: THEOREM 4.2. Let A be a cmmutative rig with i, x b which is a uit i A, ad ax xa fr each a i A. If there is a c i A such that ad (l-c i) are uits i A fr i l,..., with c i, the A(x) is Galls ver A. PROOF. We defie a mappig : A(x)A(x) by u(x) cx ad s([.alxl [ai(cx) i. The e ca check that (A(x)) A ad that s is a autmrphlsm f A(x) f rder i (fr l-c are uits i A fr i 1,2,...,-l). Mrever, sice (l-c) is a uit i A, (x-u(x)) x-cx (l-c)x is als a uit (fr x is als a uit). Therefre, A(x) is Galis ver A with a Galis grup {i,,... }([7], Prpsiti 1.2, p. 80).

7 FREE RING EXTENSIONS 709 As give i Therem 3.3, if B is Galis ver A, B(x) is a Azumaya A-algebra. We are gig t shw the existece f a splittig rig fr the Azumaya A-algebra B(x). THEOREM 4.3. Let B(x) be a Azumaya A-algebra with b ad as uits i A. The A(x) is a splittig rig fr B(x). Mrever, if is a pwer f 2, the splittig rig A(x) is a chai f Galis extesis f degree 2, ad if c ad (l-c i) are uits i A where c i, the A(x) is Galls ver A. -l.r-i i^ PROOF. Sice b ad are uits i A, the elemet u (b) ILl=0 ) x ux satisfies the equatis: ua au fr each a i A(x) ad (b)-l(xlx) i. Hece A(x) is a separable A-algebra. Mrever, e ca shw directly that A(x) is a maximal subcmmutative rig f B(x) by shwig that the cmmutat f A(x) i B(x) is A(x). Thus A(x) is a splittig rig fr B(x) ([7],Therem 5.5, p. 64). The ther results f the therem are csequeces f Therems 4.1 ad 4.2. Therem 4.1 is a geeralizati f Therem 4.2 i [4] fr quadratic free rig extesis, while Therem 4.3 prves the existece f a splittig rig fr B(x), ther tha B whe B is cmmutative ([2], Prpsiti i.i ad [5], Therem 3.2). REFERENCES i. NAGAHARA, T. ad KISHIMOTO, K. O Free Cyclic Extesis f Rigs, Math. J. Okayama Uiv. (1978), PARIMALA, S. ad SRIDHAKAN, R. Prjective Mdules ver Quaterl Algebras, J. Pure Appl. Algebra 9 (1977), KISHIMOTO, K. A Classificati f Free Extesis f Rigs, Math. J. Okayama Uiv. 181 (1976), SZET0, G. O Geeralized Quateri Algebras, Iterat. J. Math. Math. Sci. 2 (1980), SZETO, G. A Characterizati f a Cyclic Galis Extesi f Cmmutative Rigs, J. Pure Appl. Algebra I_6 (1980), AUSLANDER, M. ad GOLDMAN, O. The Brauer Grup f a Cmmutative Rig, Tras. Amer. Math. Sc. 97 (1960), DeMEYER, F. ad INGRAHAM, E. Separable Algebras ver Cmmutative Rigs, Sprlger-Verlag- Berli-Heldelberg-New Yrk, CHASE, S., HARRISON, D. ad ROSENBERG, A. Galls Thery ad Galis Chmlgy f Cmmutative Rigs, Mere. Amer. Math. Sc. 5 2 (1965). 9. MIYASHITA, Y. Fiite Outer Galis Thery f N-cmmutative Rigs, J. Fac. Scl. Hkkald Uiv. Ser..I, 1 9 (1966),

8 Budary Value Prblems Special Issue Sigular Budary Value Prblems fr Ordiary Differetial Equatis Call fr Papers The purpse f this special issue is t study sigular budary value prblems arisig i differetial equatis ad dyamical systems. Survey articles dealig with iteractis betwee differet fields, applicatis, ad appraches f budary value prblems ad sigular prblems are welcme. This Special Issue will fcus ay type f sigularities that appear i the study f budary value prblems. It icludes: Thery ad methds Mathematical Mdels Egieerig applicatis Bilgical applicatis Medical Applicatis Fiace applicatis Numerical ad simulati applicatis Cmpstela, Satiag de Cmpstela 15782, Spai; juajse.iet.rig@usc.es Guest Editr Dal O Rega, Departmet f Mathematics, Natial Uiversity f Irelad, Galway, Irelad; dal.rega@uigalway.ie Befre submissi authrs shuld carefully read ver the jural s Authr Guidelies, which are lcated at Authrs shuld fllw the Budary Value Prblems mauscript frmat described at the jural site Articles published i this Special Issue shall be subject t a reduced Article Prcessig Charge f C200 per article. Prspective authrs shuld submit a electric cpy f their cmplete mauscript thrugh the jural Mauscript Trackig System at accrdig t the fllwig timetable: Mauscript Due May 1, 2009 First Rud f Reviews August 1, 2009 Publicati Date Nvember 1, 2009 Lead Guest Editr Jua J. Niet, Departamet de Aálisis Matemátic, Facultad de Matemáticas, Uiversidad de Satiag de Hidawi Publishig Crprati