Publicacions
Ma emá iques,
Vol
36
(1992),
359-367
.
A
bs ac
NON-COMMUTATIVE
SEPARABILITY
AND
GROUP
ACTIONS
RICARDO
ALFARO
*
Dedica ed
o
he
memo y
o
Pe e
Menal
We
gi e
condi ions
o
he
skew
g oup
ing
S
*
G
o
be
s ongly
sepa able
and
H-sepa able
o e
he
ing
S
.
In
pa icula
we show
ha
he
H-sepa abili y
is
equi alen
o
S
being
cen al
Galois
ex ension
.
We
also
look
in o
he H-sepa abili y
o
he
ing
S
o e
he
ixed
sub ing
R
unde
a
ai h ul
ac ion
o
a
g oup
G
.
We
show
ha
such
a
chain
:
S
*
G
H-sepa able
o e
S
and
S
H-sepa able
o e
R
canno
occu ,
and
ha
he
cen alize o
R
in
S
is
an
Azumaya
algeb a
in
he
p esen e
o
a
cen al
elemen
o
ace
one
.
In [A]
we
in oduced
he
concep
o
sub ing-Galois
ex ensions
as
a
gene aliza ion
o
cen al
Galois
ex ensions
and
gi e
a
gene aliza ion
o
he
co espondence
heo em
gi en
by
DeMeye
in
[D]
and
Sze o
in
[SM]
.
Simila
co espondence
heo ems
we e
gi en
by
Sugano
in
[S]
using
H-
sepa abili y
.
Sepa abili y
o
non-commu a i e
ings
was
in oduced
by
Hi a a,
and
he
no ions
o
H-sepa abili y
and
"s ong"
sepa abili y
we e
in oduced
by
Hi a a
in
[HI]
and
MacMahon
and
Mewbo n
in
[MM]
espec i ely
.
S ong
sepa abili y
is
a
weake
no ion
han
H-sepa abili y,
bu
bo h
a e
special
cases
o
he
gene al
no ion
o
sepa abili y
o
ing
ex ensions
.
In
he
case
o
g oup
ac ions
we
p esen he e
condi ions
o
s ong
and
H-sepa abili y
o
skew g oup
ings
and
in
pa icula
we
show
ha
he
skew g oup
ing
S
*
G
is
H-sepa able
o e
S
i
and
only
i
S
is
a
cen al
Galois
ex ension
.
Fu he mo e,
in
his
case
S*G
is
a
Z(S)-Galois
*Pa ially
suppo ed
by a
g an
om
he
Facul y
De elopmen
Fund
o
he
Uni-
e si y
o
Michigan-Flin
and a
ellowship
om
he
Cen e
de
Rece ca
Ma ema ica,
Ba celona,
Spain
.
360

R
.
ALFARO
ex ension
(in
he
e minology
o [A]),
allowing us
o
exp ess
S
=
Z(S)R
and
S*G
=
Z(S)I
whe e
I
is
he
algeb a
o
G-cen al
unc ions
.
We
hen
s udy
he
sepa abili y
o
CS(R)
o e
i s
ixed
sub ing
and
gi e
condi ions
o
S
o
be Cs(R)-Galois
.
All
ings
he e
a e
associa i e
and
Na e
a
uni y
elemen
1
.
Z(R)
will
deno e
he cen e
o
a
ing
R,
and
CA(B)
will
deno e
he
"cen alize
o
B
in
A",
Le
.
he
elemen s
o
he
ing
A
which
commu e
wi h
all
he
elemen s
o
he
sub ing
B
o
A
.
1
.
De ini ions
and
No a ions
Le
B
be
a
sub ing
o
a
ing
A
wi h
1
.
The
ex ension
B
C
A
is
called
sepa able
(o
A
is
sepa able
o e
B)
i
any
o
he
ollowing
equi alen
condi ions
is
sa is ied
:
1)
The
mul iplica ion
mapp
:
A
®
B
A
-->
A
spli s
as
an (A
-
A)-
bimodule
map
.
2)
The e
exis s
an
elemen
e
E
A®
B
A
(called
a
sepa abili y
elemen ),
such
ha
ne
=
ea

o
all
a
E
A
and
p¿(e)
=
1
.
The
ing
A
is
said
o
be
s ongly
sepa able
o e
B
i
A®
B
A=
K
®
L
as
(A
-
A)-bimodules,
whe e
HoMA,A
(K,
A)
=
0
and
L
(D
II
-
An
o
some
(A
-
A)-bimodules
K,
L,
H
and
some
posi i e in ege
n
.
In
case
K
=
0
we
say
ha
A
is
H-sepa able
o e
B
.
S ongly
sepa able ex ensions
a e
sepa able
bu
he
con e se
is
alse,
see
[MM]
.
The e
is
an
equi alen
de ini ion
o
his
kinds
o
sepa abili y
in
e ms
o
he
na u al
(A
-
A)
-bimodule
map
cp
:
A
®
B
A
-
Hom(0~,
Aj
whe e
~o(a
®
b)(x)
=
axb,
C
is
he
cen e
o
A
and
A
is
he
cen alize o
B
in
A,
CA
(B)
.
The
ing
A
is
s ongly
sepa able
o e
B
i
and
only
i
A,
is
ini ely
gene a ed
p ojec i e
C-module
and cp
is
an
spli
epimo phism
.
Simila ly,
A
is
H-sepa able
o e
B
i
and
only
i
0,
is
ini ely
gene a ed
p ojec i e
C-module
and
cp is
an
isomo phism
.
Fo
de ails
see
[HI]
and
[MM]
.
Now
le 's
conside
g oup
ac ions
.
Le
S
be
a
ing
wi h
1,
le
G
be
a
ini e
g oup
ac ing
ai h ully
as
au omo phisms
o
S
and
le
R=
S
G
be
he
ixed ing
unde
G
.
W i ing
g( )
=
9
,
he
skew g oup
ing
S
*
G
is
he
ee
le
S-module
wi h
basis
he
elemen s
o
G
and
mul iplica ion
gi en
by
he
ule
gs
=
9sg
o
all
s
E
S
and
g E
G
.
Deno e by
-
he
elemen
E
gE
S
*
G
.
The
ac ion
o
G
on
S
is
said
o
be
G-Galois
i
9EG
S
is
ini ely
gene a ed
p ojec i e
igh
R-module
and
he
na u al
map
0
:
S
*
G
-+
EndRS
gi en
by O( g)(x)
=
(
9
x)
is
a
ing
isomo phism
;
o
equi alen ly,
he e
exis
elemen s
ai,
bi
(called a
G-Galois
basis)
such
NON-COMMUTATIVE
SEPARABILITY

36
1
ha
Y"
al
g
bi
=
1i
g
=
1
and
he
sum
is
0
i
g
=~
1
(Le
.,
Si S
=
S
*
G)
.
i
The
" ace
map",
:
S
-->
R
is
gi en
by
(x)
_

gx
which
is
an
geG
(R
-
R)-bimodule
homomo phism
.
Le
T
be
a
G-s able sub ing
o
S
( ha
is
g
E
T
o
all
E
T,
g
E
G),
we
say
ha
S
is
a
T-Galois ex ension
o
R
i
he
ac ion
o
G
on
T
is
G-Galois
.
Fo
de ails
and
p ope ies,
see
[A]
.
I
X
is
a
subse
o S,
le
I(X)
=
{g
E
G/
gx
=
x
bx
E
X}
be
he
"ine ia
g oup"
o
X,
(I(X)
is
always
a
subg oup
o
G)
.
2
.
Sepa abili y
and
skew
g oup
ings
In
[MS,
heo ems
2
.2
and
2
.3]
i is
shown
ha
i
S
is
a
simple
ing,
G
a
ini e
ou e
g oup
o
au omo phisms
o
S
and
F
=
I(Z(S)),
hen
S*
G
is
H-sepa able
o e
S
*
F
and
S
*
G
is
H-sepa able
o e
S
i
and
only
i
F
is
i ial
.
Bu
in his
case
S
*
G
is
simple
and
hence
he ac ion
o
G
on
S
is
G-Galois
.
We'll
gi e
a
gene al
esul
ela ing
G-Galois
ac ions
wi h
s ong
and
H-sepa abili y
.
Le
D
=
CS*C(S)
and
C
=
Z
(S
*
G)
.
The
ac ion
o
G
on
S
induces
a
ai h ul
ac ion
o
G
on
S
*
G
ia
conjuga ion,
ga
=
gag
-1
o
a
E
S
*
G
;
and
G
also ac s
on
D
.
Le
M
be
he
iné ia
g oup
o
D,
hus
G/M
ac s
ai h ully
on
D
by
h
a
=
9
a
o
any
g
E
h
.
Lemma
2
.1
.
D
G
=
DGIM
=
C
.
P oo
.
The
i s
equali y
is
ob ious
since
M
is
he
ine ia
g oup
o
D
.
Now
le
a
E
D
G
,
hen
ag
=
ga

dg
E
G
and
by
de ini ion
o
D,
sa
=
as

b's
E
S
;
hence
a
E
C
.
Con e sely,
i
a
E
C,
ag
=
ga

dg
E
G
and
hence
a
E
D
G
,
(is
clea
ha
C
C
D)
.
Theo em
2
.2
.
Le
M
be he
ine ia
g oup
o
D
=
CS*G(S)
and
le
C
be he
cen e
o
S
*
G
.
Assumme
he e
is
a
cen al
elemen
w
in
S
wi h
m(w)
=
1
.
I
D
is
G/M-Galois
o e
C,
hen
S
*
G
is
s ongly
sepa able
o e
S
.
P oo
.
Le
cp
:
S
*
G
®
s
S
*
G
-~
Hom(Dc,
S
*
Gc)
be
he
na u al
(S
*
G
-S
*
G)-bimodule
map,
and
le {ai, bi}
be
a
G/M-Galois
basis
o
D
o e
C
;
hen
de ine
he
maps
i
by
i(x)
=
,/,,(bix),
hus
i
E
Hom(DC,CC)
and
{ai,
i}
o m
a
dual
p ojec i e
basis
o
D
o e
C
.
Fi s

we
show
ha

{ i}

is

a
basis

o
Hom(DC,
S
*
Gc)

as
(S
*
G
-S
*
G)-bimodule
.

Fo ,
le
a
E
D,

E
Hom(D
c
,
S
*
Gc),
36
2

R
.
ALFARO
hen
(a)
=

ai i(a)

_

(ai) i(a)
_

i(a) (ai)
;
hus
i

i

i
=
E
(aá) i
=
E
i
(ai)
.
Now
we
p o e
ha
cp is
an epimo phism
.
No e
ha
W(g
®
g')
(a)
=
gc ig
-1
,
hus
cp(g
(9
g
-1
)
ac s
as
g
E
G/M
on
D
and
cp(g
(9
g
-1
)
=
~
o(h
®
h
-1
)whene e
g
=
h
in
G/M(*)
.
Choose
{h1,
...
,
h
p} a
ans e sal
o
M
in
G,
hen
j(x)
=
.IM
(bjx)
_
1
:
h¡(bjx)
=
L
.
hibjhix
h
¡
h¡
_

hi
bj<P(hi
(9
hi
1
)(x)
=
E
;P(hibj
®
hi
1
)(x)
.
The e o e
j
E
Im(
;P)
and
hence
cP is
epic
.
No ice
ha
he
exp ession
o
j
abo e
is
independen
o
he
choice
o
he
ans e sal
o
M
in
G
by
(*)
.
I
is
only
le
o
show
ha
cp
spli s
as
(S
*
G
-S
*
G)-bimodule
homomo phism
.
Le
M
be
gi en
by
he
se
{m1,
. . . ,
m
q}
and
le
l
k
=
2,3
himjwbk
®
(himj)-1
E
S
*
G®,
S
S
*
G
.
Then
'P(1k)
=
m
'w
h¡m
'bk
;P(himj
®
(himj)-1)

and
by
(*)
mjw
hi
bkW(hi(9hi
1)i
h¡-,
w
I
~o
(hibk
®
h%
1~
=
k
.
Hence
wemay
de ine
he
map
0
:
Hom(DC,
S
*
Gc)
-
S
*
G®,
S
*
G
by
linea i y
wi h
0( k)
=
lk
.
To
show
ha
0
is
an
(S
*
G
-S
*
G)-bimodule
map,
we
need
o
show
alk
=
lka
o
all
aE
S
*
G
.
Le
E
S,
since
bk
E
D
and
w
is
cen al
in
S
we
ha e
:
and
i
g
E
G,
we
ha e
:
glk
=
lk
=
57 (him
j
)wbk
®
(himj)-1
?,j
Y~
(himj)(h
i
m,
)
-l
wbk
®
(himj)-1
?j
217
him
j
wbk
®
(him')-l (h¡mj)-1
himjwbk
®
(himj)-'
=
lk ,
ghimjwbk
®
(himj)
-1
=
>~(
ghi)mjwbk
®
((ghi)mj)-1g,
1,7

áj
NON-COMMUTATIVE
SEPARABILITY

36
3
bu
{ghi}
is
ano he
ans e sal
o
M
in
G,
hence
by
(*)
glk
=
lkg and
he e o e
0
is
an
(S
*
G
-S
*
G)-bimodule
map
.
We
hen ha e
~0
Cj
:
(ak) k
l

=
~o
(
~
k

k

(ak) k
(ak)W(lk)

(ak) k
=
.
and
so
0
spli s
cp
.
k
Now
we
wan
o
show an
equi alen
condi ion
o
he
skew
g oup
ing
S
*
G
o be
H-sepa able
o e
S
.
We
s a
by
gi ing
some
no a ion
and
some
neccesa y
condi ions
assuming
all
he
no a ion
as in
heo em
2
.2
.
Fo
e e y
g
E
G
de ine
O
g
=
{
E
S/
gs
=
s

ds
E
S}
.
I
4'g
=,L
0

g
is
said
o be
w-inne , and
i
O
g
=
0
o
e e y
g
z/~
1
G
is
said
o
be
w-ou e
.
I
is
no
di icul
o
see
ha
D
=
1
:
Ogg
.
gEG
Fo he
p oo
o
he
main
heo em
we
will
need
a
esul
ha
appea s
in
[A],
and
we
ep oduce
he e
o
comple eness
.
P oposi ion
2
.3
.
([A,
p op
.
3
.3])
Assume
S*G
is
H-sepa able
o e
S
.
Then
G
is
w-ou e
and
D
=
Z(S)
.
P oo
.
-
Since
S
*
G
-
E
®
(So
g)
as
S-S-bimodules,
C,
(D)
=
S
gEG
by
[S,
p oposi ion
1
.3]
.
Hence
Z(D)
=
C,,
G(D)
n
D
C_
S
and
he e o e
C
C_
Z(D)
C_
Z(S)
.
Now
le
g
E
Wg,
so
x
=
g
g E
D,
and
hence
GI
M
(x)
=

1
:

h
g
gh
-1
=

1
:

h g
hgh
-1
E
C
C_
S
.
Thus
h
g
=
0
heGIM hEGIM
i
hgh-1
0
1,
his
is i
g
z/~
1
and
so
g
=
0
i
g
z,~=
1
.
The e o e
Y'g
=
0
i
g
:~É
1,
and
so
G
is
w-ou e
.
By
he
commen
abo e
D=
~
1
1,
so
D=
Z(S)
.
Theo em
2
.4
.
Le
D,
M,
C,
S,
G
and
w
as in
heo em
2
.2
.
D
is
G-
Galois
o e
C
and
M
is
i ial
i
and
only
i
S
*
G
is
H-sepa able
o e
S
.
P oo
. .
(=~)
Assume
he
same
no a ion as
in
he
p oo
o
heo em
2
.2
;
hibk
®
h%
1 ,
and
hence
so
now
we
ha e
lk
=
h
i
wb
k
®
(h
i
)
-
I
=
cp(1 (9 1)
=
1
:
cp(1 (S 1)
(ak)lk
=
E
a
k

k
hibk
1
h
i
®h2
1
=1®1
.
hibk
®
h2
I

36
4

R
.
ALFARO
Thus
0
-
cp
=
ids*co
S
s*G
and
(p is
an isomo phism
.
(~-=)
Assume
m
E
M
and
n
E D,
hen
cp(m
®
m
-1
)(a)
=
mam
-1
=
a =
cp(1
®
1)
(a),
bu
cp is
an
isomo phism,
hence
M
=
1
.
Now
we
will
show
D
is
G-Galois
o e
C
.
By
p oposi ion
2
.3
D
is
commu a i e,
and by
[S,
p oposi ion
1
.3]
D
is
a
sepa able
C-algeb a
.
Assume
ha
he e
exis s
a
non
ze o
idempo en
eE
D
and a
pai
h0gE
G
such
ha
9
xe
=
h
xe
o
all
xE
D
.
I
we
le
e'
=
9
e,
we
ha e
e' 7~
0
and
xe'
=
9-lh
xe'
=
e'
s
-lh
x
.
Bu
G
is
w-ou e ,
hence
g
-1
h
=
1,
hus
g
=
h,
a
con adic ion
.
The e o e
D
is
G-Galois
o e
C
by
[DI,
p oposi ion
III
.
1
.2]
.
I
S
is
a
simple
ing
and
G
is
ou e ,
hen
Z(S)
is
a
ield,
and
hence
G/M
is
G/M-Galois
o e
Z(S)
whe e
M
=
I(Z(S))
.
The e o e
applying
he p e ious
heo ems
we
ob ain
an
imp o emen
o
[MS,
Theo em
2
.3
and Theo em
2
.2,ii)]
Co olla y
2
.5
.
Le
S
be
a
simple
ing
and
G
be
ou e
.
i)
I
3w
E
Z(S)
such
ha
m(w)
=
1,
hen
S
*
G
is
s ongly
sepa able
o e
S
.
ii)
S
*
G
is
H-sepa able
o e
S
i
and
only
i
M
=
1
.
We
can
see
now
a
ela ionship
be ween
H-sepa abili y
and
T-Galois
ex ensions
in
he
ollowing
co olla ies
:
Co olla y
2
.6
.
S
*
G
is
H-sepa able
o e
S
i
and
only
i
S
is
a
cen al
Galois
ex ension o
R
.
P oo
.
(~) 9
ai,
bi
E
Z(S)
such ha

E
a
i
7
G
b
i
=
1,
bu
Z(S)
C_
Cs
*
G(S)
=D
and
D
is
G-in a ian ,
hence
D
is
G-Galois
o e
DG=C
and
by
heo em
2
.4
S
*
G
is
H-sepa able
o e
S
.
(=)
Ob ious
om
he
heo em
2
.4
and
p oposi ion
2
.3
.
The
case o
commu a i e
ings
is
now
de e mined
:
Co olla y
2
.7
.
Le
S
be
a
commu a i e
ing
.
S
*
G
is
H-sepa able
o e
S
i
and
only
i
S
is
G-Galois
o e
R
.
Conside
again
he
ac ion
o
G
on
S
*
G
by
conjuga ion
.
I
ollows
ha
he
cen aliza
o
G
in
S
*
G
is
p ecisely
equal
o
he
ixed
ing
(S
*
G)
G
=
I,
which
in
he
language
o
C*-algebas
is
callad
he
algeb a
o
G-cen al
unc ions,
(see
[OP])
.
Hence
we
ob ain
:
P oposi ion
2
.8
.
Le
S
*
G
be
H-sepa able
o e
S
.
Then
S
*
G
is
a
Z(S)-Galois
ex ension
o
I
and
he e o e
S
*
G
=
Z(S)I
.
NON-COMMUTATIVE
SEPARABILITY

36
5
3
.
H-sepa abili y
and
ixed
ing
Now
we
s udy
some
neccesa y
condi ions
o
he
ing
S
o
be
H-
sepa able
o e
he
ixed
ing
R
.
The
cen alize
o
R
in
S
will
be
deno ed
by
E
and
all
he
no a ion
om
Sec ion
2
will
be
assumed
.
Le
X
be a
G-in a ian
subse
o
S
.
I
can
be
easily
seen
ha
CS(X
)
is
a
G-in a ian
sub ing
o
S
and
hus
G
ac s
on
i
.
Flz he mo e
we
ha e
ha
(CS(X))G
=
CR(X)
.
Hence,
i
we
ake
X
=R
we
ge
he
ollowing
ela ion
:
EG=
Z(R)
C
Z(E)
.
On
he
o he
hand
i is
ob ious
ha
Z(S)
C
Z(E)
.
P oposi ion
3
.1
.
Le
S
be
H-sepa able
o e
R
.
Then
:
1)
G
is
w-inne
.
2)
R
=
CS(E)
3)
E
G
=
Z(R)
=
Z(E)
P oo
.
1)
Recall
ha
4'9
=
{
E
S/
gs
=
s

ds
E
S}
.
Conside
he
(S
-
S)-bimodule
Sg
.
Then
Eg=
Cs,~g(R)
and Ogg
=
Csg(S),
he e o e
we
ge
Eg=
E
OZ(S)
Ogg
and
hence
4'g =~
0
.
2)
I
is
clea
ha
R
C_
CS(E)
.
Now,
le
E
CS(E)
and
le
g
E
G
.
We
can
see g as
an
elemen
o
HOMR-R
(S,
S)
which
is
isomo phic
o
E%(S)
E
by
[H2,
p oposi ion
4
.7]
.
Thus
he e
exis s
elemen s
di,
el
E
E
such
ha
gx
=J
:i
dixei
o
all
x
E
S,
and
he e o e
g
di ei
=
j
:
i
diei
=
;
so
G
R
.
3)
By
he
commen s
abo e,
i
is
only
neccesa y
o
show
he
second
equali y
.
Bu ,
by
pa
2)
we ha e
:
Z(R)
=R
n
Cs(R)
=
R
n
E
_
Cs(E)
n
E=
Z(E)
.
a
Rema k
.
No e
ha
in
p oposi ion
2
.3
we
showed
ha
i
he
skew
g oup
ing
S
*
G
is
H-sepa able
o e
he
base
ing
S,
hen
he ac ion
o
G
mus
be
w-ou e
.
He e
we
ob ain
he
opposi e
condi ion,
i
he
ing
S
is
H-sepa able
o e
he
ixed
ing
R,
he ac ion
o
G
mus be
w-
inne
.
The e o e
we
canno
ha e a
"chain"
o
H-sepa abble
ex ensions
in
ai h ul
g oup
ac ions
.
P oposi ion
3
.2
.
Le
S
be
H-sepa able
o e
he
ixed
ing
R
and
assume
he e
exis s
a cen al
elemen
in
S
o
ace
one
.
Then
E
is
sepa able
o e
Z(S) and
H-sepa able
o e
E
G
(so
E
is
an
Azumaya
algeb a)
.
P oo
. .
The
exis en e
o
a
cen al
elemen
o
ace
1
makes
he
ace
map
:
S
-4
R
spli
as
a
(R
-
R)-bimodule
map
.
Hence
R
is
a
di ec
summand
o
S
as
(R
-
R)-bimodules
and by
[S,
p oposi ion
1
.3]
E
is
36
6

R
.
ALFARO
sepa able o e
Z(S)
.
Fu he mo e,
since
Z(S)
C_
Z(E),
he
heo em
o
Azumaya
o
sepa able
ex ension
o e
commu a i e
ings
implies
ha
E
is
sepa able
o e
i s
cen a
Z(E)
and
Z(E)
is
sepa able
o e
Z(S)
.
The e o e,
E
is
H-sepa able
o e
Z(E), which
by
p oposi ion
3
.1
is
equal
o
he
ixed
sub ing
EG
.
The
ac ion
o
G
on
S
induces
an
ac ion
on E,
bu
we
need
o
conside
he
ine ia
subg oup
K
=
I(E)
.
In his
way
G/K
ac s
ai h ully
on
E
.
We
now
desc ibe
condi ions
o
E
o
be
a
Galois ex ension
o
EG
.
P oposi ion
3
.3
.
g E
K
i
and
only
i
Og
C
Z(E)
.
P oo£
Since
Og
C_
E
he
neccesa y
condi ion
is
ob ious
.
Now
le
a
E
Og
C
Z(E)
;
hen
a(
9
x
-
x)
=
0
o
all
x E
E
and
he e o e
gx
=
x
o allxEE
.
Theo em
3
.4
.
Le
S
be
H-sepa able
o e
R
and assume
he e
is
a
cen al
elemen
o
ace 1
.
S
is
an
E-Galois
ex ension
o
R
i
and
only
i
C
=
E
G and
K
is
i ial
.
P oo
.
(=)
By
de ini ion
o
E-Galois
ex ension,
K
is
i ial
and
he
ac ion
o
G
on
E
is
G-Galois,
mo eo e
by
p oposi ion
3
.2
E
is
H-
sepa able o e
E
G
.
Fu he mo e,
by
[S2],
E
_

Og
is
a
di ec
sum
and
9
Og
=
Cx
g
,
hus
p oposi ion
3
.3
implies
ha
Z(E)
=
C,
so p oposi ion
3
.1 gi es
us
he
esul
.
(~)
Since
K
is
i ial
and
he
ixed
elemen s
in
E
coincide
exac ly
wi h
he
cen al
elemen s
we
ha e
ha
he
sum

~9
is
di ec
;
mo eo e
9
in his
case
E
=CE
(E
G
)
and
E
G
=
Z(E)
gi ing
us
CE(EG
)
equal
o
he
di ec
sum
o he
co esponden
09
.
Thus
by
[S2,
heo em
1
.2]
he
ac ion
o
G
on
E
is
G-Galois
.
Re e ences
[A]
ALFARO
R
.,
T-Galois
Ex ensions
on
Rings
and
a
submodule
co -
espondence,
Comm
.
i
n
Algeb a,
o
appea
.
[D]
DEMEYER
F
.,
Some
no es
in
he
gene al
Galois
Theo y
o
ings,
Osaka J
.
Ma h
.
2
(1965),
117-127
.
[DI]
DEMEYER
F
.
AND
INGRAHAM
E
.,
"Sepa able
Algeb as o e
Commu a i e
Rings,"
Zec u e
No es
in
Ma hema ics
181,
Sp inge -Ve lag,
1971
.
NON-COMMUTATIVE
SEPARABILITY

36
7
[HI]
HIRATA
K
.,
Some
ypes
o
sepa able ex ension
o
ings,
Nagoya
Ma h
.
J
.
33
(1968),
107-115
.
[H2]
HIRATA
K
.,
Sepa able
ex ensions
and
cen alize s
o
ings,
Nagoya
Ma h
.
J
.
35
(1969),
31-45
.
[MS]
MCMAHON
E
.
AND
SHAPIRO
J
.,
On
s ong
and
H-sepa abili y
in
o dina y
and
Skew
g oup
ings,
Hous on
J
.
Ma h
.
15,
no
.
3
(1989)
.
[MM]
MCMAHON
E
.
AND
MEWBORN
A
.
C
.,
Sepa able
ex ensions
o
non-commu a i e
ings,
Hokkaido
Ma h
.
J
.
XIII, 1
(1984)
.
[OP]
OSTERBURG
J
.
AND
PELIGRAD
C
.,
A
S ong
Connes
Spec um
o ini e
g oup
ac ions
o
simple
ings,
J
.
Algeb a,
o
appea
.
[S]
SUGANo
K
.,
On
cen alize s in
sepa able
ex ensions,
Osaka J
.
Ma h
.
7
(1970),
29-40
.
[S2]
SUGANo
K
.,
On
a
special
ype
o
Galois
ex ensions,
Hokkaido
Ma h
.
Jou nal9
(1980),
123-128
.
[SM]
SZETO
G
.
AND
MA
L
.,
On
cen e -Galois
ex ensions o e
a
ing,
Glasnik
Ma ema icki
24
(44)
(1989),
11-16
.
Uni e si y
o
Michigan-Flin
MI
48502
U
.S
.A
.
Rebu
el
25
de
No emb e
de
1991