Note on automorphisms in separable extension

Transcription

1 Hokkaido Mathematical Journal Vol 9 (1980) p Note on automorphms in separable extension non commutative ring By Kozo SUGANO (Received April ) Preliminaries All definitions and terminologies in th paper are the same as those in the same author s papers [8] [11] and [13] So shall a ring with an identity 1 a subring which contains 1 C the center C the center and { =V_{}()= x\in xr=rx all r\in } an H- separable extension \otimes_{} if a -direct summand some finite - direct sum copies In th case a separable extension i e map \pi \otimes_{} to such that \pi(x\otimes y)=xy x y\in splits as --map As the fundamental properties H-separable extension see [4] [5] and [12] In [11] and [13] the author showed that in case a simple artinean ring an H separable extension if and only if _{-} an inner Galo extension It well known that in th case every automorphm which fixes all elements an inner automorphm In th paper we will generalize th theorem to the case ordinal H- separable extensions (Theorem 2) We will also show that every G-Galo extension such that all elements G are inner automorphms an H- separable extension (Theorem 3) For a tw0-sided -module M we denote C-submodule {m\in M xm=mx all } x\in M^{} by Then H-separable over \otimes_{c}m^{}\cong M^{} if and only if by (d\otimes marrow dm) (see Theorem 1 2 [8]) every two sided -module M We will use th theorem very ten throughout th paper For a ring we denote the Jacobson radical by J() We will also study in \S 3 in what case J()= J()=J() and holds when J()=J()\cap H separable over 1 Automorphms in H-separable extensions The first result a supplement Theorem 2 [5] THEOREM 1 Let an H separable extension Then every ring endomorphm which fifixes all elements an automorphm and fifixes all elements V_{A}(V_{A}())

2 _{} \overline{} which ^{-} by Then fixes by Note on automorphms in separable extension non commutative ring 269 PROOF Let an arbitrary ring endomorphm with (r)=r all Then r\in- \in Hom(_{}_{ }_{})\cong\otimes_{c}^{0} (see (1 5) [12]) Hence there \sum d_{i}\otimes e_{i}^{0}\in\otimes_{c}^{0} exts such that all (x)=\sum d_{i}xe_{i} x\in Then any r\in V_{A}() (r)=r\sum d_{i}e_{i}=r since (1)=\sum d_{i}e_{i}=1 Thus fixes all elements V_{}() C\subset V_{}() Then fixes all elements C since Then by Theorem 2 (b) [5] an automorphm THEOREM 2 Let an H separable extension \overline{}= and let /J()\overline{}=/J()\cap ^{-}=V_{I}\overline{4}(\overline{}) and Then if 11 artinean and if mapped onto the natural map every automorphm which fifixes all elements an inner automorphm In order to prove th theorem we need the following PROPOSITION 1 Let a separable extension \alpha and an ideal which contained in Let J() an automorphm which fifixes all elements Then if induces the identity automorphm \overline{}\overline{}(\overline{x})=\overline{(}x) all x\in \overline{}=/\alpha an inner automorphm where and in \overline{} \overline{x}=x+a x\in PROOF Let \delta(x)=(x)-x Then x\in \delta a -derivation to a -module - Q where the right -module structure a defined by a\cdot x=a(x) and Then by Satz 4 2 [2] a\in\alpha x\in \delta an inner derivation and there exts a\in 0 such that (x)-x=xa-a(x)(=\delta(x)) x\in all Hence (1+a)(x)=x(1+a) But since 1+a a unit a\in J() Theree an inner automorphm PROPOSITION 2 Let a two sided simple ring [not necessarily artinean) and an H-separable extension some subring Then every automorphm which fifixes all elements an inner automorphm PROOF Let an automorphm with (x)=x x\in all Let a -bimodule defined by the following way; - _{}= as left -module _{} but right -module structure defined by x\cdot y=x(y) y\in x J_{}= Let { a\in xa=a(x) any } Then clearly x\in (_{})^{}=J_{}\subseteq and (_{})^{}= On the other hand =(_{})^{}\cong\otimes_{c}(_{})^{}=\otimes_{c}j_{} since an H- J_{} separable extension Then since C a field [ : CJ =1 and J_{}=Cu_{} 0\neq u_{}\in J_{} u_{} some Then clearly a two sided ideal a simple ring u_{}= u_{}\in Theree u_{} and we see that an unit Since J_{} we have all (x)=u_{}^{-1}xu_{} x\in \overline{} =V- PROOF THEOREM 2 Let and (^{-}) \overline{c} the center \overline{} By \overline{} Proposition 3 2 [13] and Theorem 1 3 [8] an H separable extension \overline{} \overline{} both and Since (J())=J() \overline{} induces an automorphm \overline{} \overline{} \overline{} fixes all elements all elements Theorem

3 \overline{} induced induced to primitive are as into which 270 K Sugano \overline{} 1 Since \overline{} \supseteq\overline{c} all central idempotents also central idempotents \overline{} \overline{}=\overline{}_{1}\oplus\overline{}_{2}\oplus\cdots\oplus\overline{}_{2} Hence if a decomposition simple rings \overline{1}=\overline{e}_{1}+\overline{e}_{2}+\cdots+\overline{e}_{n}\overline{}_{i}=\overline{}\overline{e}_{i} \overline{e}_{i} and if with idempotents the center \overline{}_{1} \oplus\overline{}_{2} \oplus\cdots\oplus\overline{}_{n} =\overline{} \overline{}_{i} =\overline{} \overline{e}_{i} \overline{}_{i} \overline{} then with ring and a subring \overline{}_{i} _{i}^{-} \overline{}_{i} each i Then clearly each an H-separable extension _{i}^{-} \overline{}_{i} \overline{} \overline{}_{i} and the restriction an automorphm fixes \overline{}_{i} \overline{}(\overline{e}_{i})=\overline{e}_{i} \overline{}_{i} all elements cause each i Theree each an _{i}^{-} V_{\overline{A}_{i}}(\overline{}_{i} ) \overline{} inner automorphm by a unit Then an inner \overline{} \overline{d} ^{-}=\sum^{\oplus}v_{\overline{}_{i}}(\overline{}_{i} ) automorphm by a unit Let such \overline{d} a unit ^{-} i e\overline{}(\overline{x})=\overline{d}^{-1}\overline{x}\overline{d} all \overline{x}\in\overline{} and d a representative \overline{d} in By assumption we can choose d from Since a unit in \overline{} we have dd =1+m some d \in and But 1+m a unit m\in J() \tau Hence d also a unit in Let an automorphm defined by \tau(x)=d(x)d^{-1} x\in \tau all Then fixes all elements since d\in \overline{\tau}(\overline{x})= \tau and we see that identity on \overline{} Then by Prop 1 an inner automorphm and also an inner automorphm 2 Relation with Galo extensions Let a ring and G a finite group automorphms ^{G}= { x\in (x)=x all } \in G Let S=(:G) the trivial crossed product \{U_{}\}_{\in and \sum_{\in G}\oplus U_{} a free -module with a free bas G} G that S= \lambda where the product defied by there exts a ring homomorphm U_{}\gamma U_{\tau}=\lambda(\gamma)U_{\tau} \lambda \gamma\in \tau\in G Then j:(:g)arrow Hom(_{} _{}) j(\lambda U_{})(x)=\lambda(x) \lambda x\in \in G Now following Tr Kanzaki [6] we say that a G-Galo extension \wedge if (1) =^{G}(2) right -finitely generated pr0- jective and (3) map j an omorphm Lemma 1 Let a G-Galo extension (1) There exts c\in C with t_{g}(c)=1 if and only if <\oplus_{}_{r} t_{g}(x)= \sum_{\in G}(x) x\in (2) Suppose furthermore C\subseteq only if \tau r<\oplus_{}_{} Then we have where then G =n a unit in C if and PROOF (1) If there exts c\in C with t_{g}(c)=1 we obtain a - - map f to defined by f(x)=xc Then we have x\in (t_{g}\circ f)(r)= t_{g}(rc)=rt_{g}(c)=r r\in all Theree \tau_{}<\oplus_{}_{} Conversely suppose (_{} _{}) <\oplus_{}_{} Then since right -finitely generated projective Hom a separable extension by Theorem 7 [10] Then S=( G)

4 Note on automorphms in separable extension non commutative ring 271 \sum\alpha_{i}\otimes\ta_{i}\in(s\otimes_{}s)^{s} a separable extension Hence there exts with \sum\alpha_{i}\ta_{i}=1 \sum\alpha_{i}\otimes\ta_{i}=\sum x_{\tau}u_{}\otimes U_{\tau}=\sum x_{\tau}-1u_{}\otimes U_{\tau}-1 We can put where \tau\subseteq G x_{\tau}\in \sum\alpha_{i}\ta_{i}=1 \sum x_{\tau}-1u_{\tau}=u_{1} and implies Hence we have \sum x_{}-1=1 and \sum x_{\tau}-1=0(\tau\neq 1) On the other hand \sum U_{\rho}\alpha_{i}\otimes\ta_{i}=\sum\alpha_{i}\otimes \ta_{i}u_{\rho} all \rho\in \rho(x_{\tau})=x_{\rho\tau\rho}-1 G implies that \tau \rho\in \sum x\alpha_{i}\otimes G and \ta_{i}=\sum\alpha_{i}\otimes\ta_{i}x x\in all x_{\tau}\in J_{\pi} implies that all \tau\in G Especially we have \rho(x_{11})=x_{\rho\rho}-1 and x_{11}\in J_{1}=C Hence we have 1= \sum x_{}-1=\sum(x_{11}) (2) follows from (1) since t_{g}(n^{-1})= \sum n^{-1}(1)=n^{-1}n=1 and implies x_{11}\in C\subseteq that t_{g}(x_{11})= \sum(x_{11})=nx_{11}=1 PROPOSITION 3 Let an H separable and G-Galo extension Then we have - (1) V_{}(V_{}())= (2) a rank n projective module where n= G (3) Following three conditions are equivalent (i) n(= G ) a unit (ii) a -direct summand - (iii) a separable C-algebra PROOF (1) By Prop 1 we see that every element G fixes all element Thus we have (=V_{}()) \underline{\subset}^{g}=- The converse inclusion =(_{})^{}\cong\otimes_{c}(_{})^{}=\otimes_{c}j_{} obvious (2) By (1 3) (4) [12] It already known that C-finitely generated projective and C a C-direct summand J_{} Hence rank one projective C-module \cong On the other hand (_{} _{ }_{})\cong(:g)^{}=(\sum^{\mp}\sim U_{})^{}=\sum_{\check{}\in G}^{+\neg}J_{} Hom Thus rank n projective C-module (3) C\underline{\subset}V_{}())= Since by (1) the equivalence (i) and (ii) follows from Lemma 1 The equivalence (ii) and (iii) follows from Prop 4 7 [3] and Corollary 1 2 [9] since V_{}(V_{}())=- But the author will repeat the pro here the convenience to readers Suppose (ii) and let p the map - to with p(1)=1 Then we have a commutative diagram -maps - all (_{}_{ }_{}) \bigotimes_{\nearrow\pi}c^{\frac{\eta}{\searrow\swarrow\varphi_{\nearrow}}hom}\backslash \nwarrow Hom\backslash (p 1_{}) arrow Hom(_{}_{ }_{}) where \eta(d\otimes e)(x)=dxe \pi(d\otimes e\in e)=de d and x\in \varphi(f)=f(1) f\in (_{}_{ }_{}) Hom and n(d)=dr(=rd) d\in r\in\wedge and n are omorphms \eta (see (1 5) [12]) Thus \pi splits as -map - Suppose (iii) Then \sum d_{i}\otimes e_{i}\in(\otimes_{c})^{} there exts with Hence we obtain map p \sum d_{i}e_{i}=1 to such that (=V_{A}()) p(x)= \sum d_{i}xe_{i} x\in all p a map -

5 272 K Sugano with p(r)=r all Thus we have r\in- (ii) As an example H-separable G-Galo extensionst we have THEOREM 3 Let a G-Galo extension Then if all elements G are inner automorphms then an H-separable extension and a free C-module rank n where n= G PROOFS For each let \in G (x)=\gamma_{}^{-1}x\gamma_{} \gamma_{} a unit such that x\in s U_{} all Note that each a -module with mulae - U_{}\lambda= (\lambda)u_{} \lambda\in each and that j a -omorphms Then each - \in G f_{} U_{} f_{}(\lambda U_{})=\lambda\gamma_{}-1 define a map to by Then since f_{}(u_{}\lambda)=f_{}((\lambda)u_{})=(\lambda\rangle \gamma_{}^{-1}=\gamma_{}^{-1}\lambda=f_{}(u_{})\lambda \lambda\in each \lambda\in f_{} each a - - omorphm Hence we have (_{} _{})=\oplus\oplus\cdots\oplus Hom (n folds) as --module Then \cong Hom(_{A}_{ }_{})=[Hom(_{} _{})]^{A}\cong[\oplus\oplus\cdots\oplus]^{A}=C\oplus C\oplus\cdots\oplus C Hence a free C-module rank n On the other hand since -finitely generated projective we have right \otimes_{}\cong\otimes_{}hom(_{ A})\equiv Hom(_{A}Hom(_{} _{}) ) \cong Hom ( (\oplus\oplus\cdots\oplus) )\cong\oplus\oplus\cdots\oplus as -module Thus - an H-separable extension REMARK In the pro Theorem 3 we see that the -omorphm to given by; (:G) - \oplus\oplus\cdots\oplus ( \lambda_{\rho} \lambda_{} \cdots \lambda_{\tau})arrow\sum_{\in G}\lambda_{}\gamma_{}U_{} (_{A}_{ A}_{})arrow On the other hand the omorphm Hom given by; farrow f(1) f\in Hom(_{A}_{ }_{}) Theree mapped C\oplus C\oplus\cdots\oplus C onto j( \sum_{\epsilon G}\oplus C\gamma_{}U_{}) = \sum_{\epsilon G}\oplus C\gamma_{} Thus we have V_{A}( )=\sum_{\epsilon G}\oplus C\gamma_{} REMARK a G-Galo extension if and only if there ext x_{l} y_{i}\in(i=12 \cdots n) such that \sum x_{i}(y_{i})=_{1} by Prop 2 4 [6] Then under the condition Theorem 3 it can directly computed that 1\otimes 1= \sum_{\in G}\gamma_{}(\sum x_{i}\otimes(y_{i})\gamma_{}^{-1}) in We call these \{\gamma_{} \otimes_{} with \gamma_{}\in \sum x_{i}\otimes(y_{i})\gamma_{}^{-1}\}_{\in G} an and \sum H-system 3 On radicals in H-separable extensions Then J()\cap- x_{i}\otimes(y_{i})\gamma_{}^{-1}\in(\otimes_{})^{} (see [5]) PROPOSITION 4 Let 11 an H-separable extension with if /J() <\oplus artinean we have J()= J()=J() and J()=

6 \overline{} Then Note on automorphms in separable extension non commutative ring 273 PROOF By Theorem 4 1 (2) [13] J()=(J()\cap)=(J()\cap) Hence we need only to show that J()=J()\cap_{q} Since =V_{A}() every J()\cap element has its quasi-inverse in Theree J()\cap\subseteq J() \overline{}=/j() Let \overline{}=/j()\cap and \overline{} Then an H separable extension \overline{r}\overline{}\overline{r}<\oplus_{\overline{i}}\overline{}_{\overline{r}} \overline{}-\overline{} \overline{}=\overline{}\oplus M \overline{} and by Prop 3 4 (1) [13] Let as - \mathfrak{l} module and \overline{}- an arbitrary left ideal \overline{}=\overline{}1\oplus L \overline{} - as left module Then and as left \overline{}-module \overline{}=(\mathfrak{l}\oplus M\mathfrak{l})\oplus L \overline{}=\mathfrak{l}\oplus(m\mathfrak{l}+l)\cap\overline{} \overline{} Thus every left ideal a \overline{}-direct summand \overline{} and we see that a semimle ring Then J(\overline{})=0 and J()\underline{\subset}J()\cap\wedge Theree we have J()=J()\cap REMARK In general J()= J()= J() and J()\cap=J() do not hold in H-separable extensions Let D a divion ring and the n\cross n-full matrix ring over D and the lower triangular matrix subring Let e_{ij} the matrix units Then it easily proved that \sum e_{i1}\otimes e_{1i}\in(\otimes_{c})^{} \sum e_{i1}e_{1i}=1 But since each i e_{i1}\in \sum e_{i1}\otimes e_{1i}\in(\otimes_{d})^{} Hence map \pi \otimes_{d} to defined by \pi(r\otimes x)=rx(r\in x\in) splits as -map Then by Prop 2 2 [9] 11 an H-separable - extension It also clear that left -finitely generated projective But J()=0 and J()\neq 0 Bee explaining some examples in which the conditions Theorem 4 holds we need some preperations The next two propositions are supplements results which have en obtained in [13] PROPOSITION 5 Let such that r<\oplus_{} (_{}) Then Hom a left -progenerator PROOF <\oplus_{r} Since (_{}) Hom a left -direct summand (_{}) Hom (_{})=\otimes_{c}<\oplus\oplus\oplus\cdots\oplus But Hom as -module - since C-finitely generated projective Hence (_{}) Hom left - finitely generated projective On the other hand in Prop 1 1 (1) [13] we (_{}) have already shown that Hom a left -generator PROPOSITION 6 Let an H separable extension Then (1) If left -cogenerator then a left -cogenerator (3) If a left PF ring then a left PF-ring (3) If left self injective then left self-injective (4) If a quasi-fronius ring then a quasi-fronius ring PROOF (3) and (4) are shown in [13] Hence we need only to show (1) But th follows from Korollar 1 (_{})\underline{\subset} (_{})<\oplus [15] since Hom Hom \oplus\oplus\cdots\oplus as left -module Since left PF-ring a ring which left self injective and a left cogenerator (see [1]) (2) follows from (1) and (3) an H separable extension

7 274 K Sugano It well known that if a left (or right) PF ring artinean /J() Theree we have COROLLARY 1 Let a left (or right) PF ring and an H-separable extension Then if a -direct summand - J()= J()= J() and J()=J()\cap References [1] G AZUMAYA: Completely faithful modules and self-injective rings Nagoya Math J 27 (1966) [2] S ELLIGER: Ur Automorphmen und Derivationen von Ringen J reine angew Math 277 (1975) [3] K HIRATA: Some types separable extensions Nagoya Math J 33 (1968) [4] K HIRATA: Separable extensions and centralizers rings Nagoya Math J 35 (1969) [5] T NAKAMOTO and K SUGANO: Note on H-separable extensions Hokkaido Math J 4 (1975) [6] T KANZAKI: On Galo extension rings Nagoya Math J 27 (1966) [7] T KANZAKI : On Galo algebra over a commutative ring Osaka J Math 2 (1965) [8] K SUGANO: Note on semimple extensions and separable extensions Osaka J Math 4 (1967) [9] K SUGANO: On centralizers in separable extensions Osaka J Math 7 (1970) [10] K SUGANO: Note on separability endomorphm rings J Fac Sci Hokkaido Univ 21 (1971) [11] K SUGANO: On some commutor theorems rings Hokkaido Math J 1 (1972) [12] K SUGANO: Separable extensions quari-fronius rings Algebra-Berichte 28 (1975) Uni-Druck Munchen [13] K SUGANO: On projective H-separable extensions Hokkaido Math J 5 (1976) [14] K SUGANO: On automorphms in separable extensions rings Proc 13th Symposium ring theory 1980 Okayama Japan [15] T ONODERA : Koendlich erzeugte Moduln unt Kogenerator Hokkaido Math J 2 (1973) Department Mathematics Hokkaido University 060 Sapporo Japan