149
Internat. J. Math. & Math. Sci.
(1991) 149-154
VOI. 14 NO.
ON GALOIS PROJECTIVE GROUP RINGS
LINJUN MA
and
GEORGE SZETO
Mathematics Departmezt
DePartment
Mathematics
Zhongshe n University
Guangzhou, P.R. China
Bradley University
Peoria, Illinois 61625 U.S.A.
(Received September 19, 1989 and in revised form December 10, 1989)
ABSTRACT.
Let
A
be a ring with
inner automorphism group of
A
I,
C
induced by
A
the center of
[U
A G’
in
A
/
and
G’
an
O in a finite
be the fixed subring of
Let
group G whose order is invertible.
G’ with
If A is a Galcis extension of A
A under the action of G’
then
Galois group G’ and C is the center of the subring
G’ is also C.
G’
if
Moreover,
A
U and the center of A
is Azumaya over C, then A is a projective group ring.
AG’u
=A
@AG’U
KEY WORDS AND PHRASES.
Central Galois extensions, projective group
rings, Azumaya algebrss.
1980
AMS SUBJECT CLASSIFICATION CODE.
I.
INTRODUCTION.
16A74, 16A16, 13B02.
Galois extensions for rings and Azumaya algebras have been inten-
F.R. DeMeyer ([I]) characterizes
a central Galois algebra with an inner Galois group in terms of an Azumasively investigated (see References).
That is, if A is a central Galois algeya projective group algebra.
{U
bra over C with an inner Galois group G’ induced by the units
algebra
group
a
is
projective
then
A
in a finite group
/
Conversely,
in G.
for 4,
where f is a factor set f(,)
if
is Azumaya over C, then it is Galois over C with an inner Ga-
CGf
G},
CGf
UUU
In the pre(I, Theorems 2 and 3).
G’ induced by
G’ (not
a
ring A
sent paper, we shall study a Galois extension A over
It will be shown that if A is Galois with
necessarily its center C).
G’
Galois group G’ and if the subring
U has center C, then A G’
G’ is also C.
G’
In addition, if A
U such that the center of A
is separable over C, then A becomes a projective group ring over A
A
(that is, the coefficient ring C in a projective group C-algeG’ as defined in I).
In this case, A is an
bra is replaced by a ring A
G’
Azumaya C-algebra
Conversely, if
U is an C-Azumaya C-algebra
A G’ Gf and is Galois over
such that U} are free over C, then A
G’ has
Our results generalize the characterizacenter C.
such that A
lois group
U
A
A
G’,
AG’Gf
A
tion for a central Galois algebra as given by F.Ro DeMeyer
AG’
([I],
Theorems
150
G. SZETO AND L. MA
2 and 3).
This paper was doric under the support from the Advanced Re-
search Institute of Zhongshan University, P.R. China, and revised under
We would like to thank the referee for
the suggestions of the referee.
his valuable comments to relate our paper to the work of T. Kanzaki.
BASIC DFFINITI ONS
Throughout, we assume that
2.
A,
Ucz
c
A /
/
agroup
in
G
A
is a ring with
the inner automorphism group of
G’
ring
in a finite group
’(a)
over a ring
RGf
UU
such that
r
in
sion
T
over
S
ai,
aaib i
aibia
’
where
in
b
bi}
UU#U-
f(,)
and
.
G
in
T
in T, i
i
for each a in T
Such a set
invertible,
G’.
I,
and
U
/ c
r
rU
and
A
G’
The projective
is a ring with an R-basis
is a ring extension
there exist elements
over S.
G
in
the center of
C
induced by the set of units
for each
R
Uf(,)
,
R,
each
Ua
a
A
whose order is
G
I,
in
for
A separable exten-
over its subring S such that
m for some integer m /
....
and
I
aib
where
is
i
is called a separable set for
T.
The
ai,
T is a separable extension T over S which is
contained in the center of T, and T is an Azumaya S-algebra if T is
a separable algebra over its center S.
T is a Galois extension over
T G with Galois group G if G is a finite automorphism group of T
and there exist elements
Yi in T, i 1,...,k for some integer k
/
and
0
for each
in
Such a set
xia(y i)
i
is
called
a
Galois set for T.
A Galois extension T over T G
Yi)
with Galois group G is called a centralized Galois extension if T G has
the same center as T.
GALCIS PROJECTIVE GROUP RINGS.
3.
Throughout, we assume that A is a ring with I, C the center of
A, G’ an inner automorphism group of A induced by
in A / c
in a finite group G whose order is invertible.
We denote the set
in A /
a for each oC’ in
’(a)
by A G’ and the projective
G’
G’
group ring over A
by A Gf where f(c%,)
for
in G.
We shall generalize the characterization of a central Galois algebra as given by F.R. DeMeyer to a non-commutative case
We begin with some properseparable S-algebra
xi’
xiY
Xl’
G.
U
a
ties of
G’
UU U-
c,
G’.
LEMMA 3I.
(I) Let
(or A G’) and U3
morphic with
C, then G’
PROOF.
G’
CU
restricted to
AG’U,
c’
A
AG’
restricted to
Since
.AG’u
(or
respectively--Then
’
on
G’
Ua.
be the subring generated by
G’ restricted to
G’
(2) If
A
is isomorphic with
A G’
U/- Io-. is
U/- I.
center of
and since o’
on A if and only if
C similarly, part (2) holds
Part (I) is clear.
LEMMA 3.2
If A is Galois over A G’ with Galois group G’
G’ G.
Uo
G
m
a
in the
is in
then
PROOF.
We first show that
/ cz in
are free cver
fact, let
0 for
in C and let {’xi,
Yi in A, i
for some integer
be a Galois set for A.
Then
aU
is iso-
has center
G’.
if and only if
’
C
C
C.
In
,m
xi(2a)-1(yi)
GALOIS PROJECTIVE GROUP RINGS
a(xi{1(yi))U
O=
be in
G’
is in
C.
’(a)
Thus there is an
cU.
Ui
Moreover,
c
the map
G’
phism, so
THEOREM 3.3.
If
roup G’
-
U]
But
G.
Let
for any
aU6
’(a)
such that
A
c
C
are free over
@’
Go
a
in
G’
C, so
U
to
is
G
w AG’u
has center
U.
clearly
C, then
A
f(g,)
U@UI$
(I/n)UU
der of
Now, let
o
in
G.
in
G, so
AG’U
is a separable element for
which is
{U} are
CGf of A
free over
G
CGf
Ut
U,
A
where
G
is invertiis the or-
n
where
C.
is a separable projective group algebra over
t
A G’
in
Hence rt
tr
is in the center of
(for
G
tUck.
A
r
r
rU
Then
CGf.
On the other hand, for any
A
CG f
Since
with Galois
Since the order of
be an element in the center of
r
each
each
Hence
G.
G.
in
G’
A U
{U}.
,
.
a group homomor-
G’
A
G’
and
A
As given in the proof of Lemma 3.2,
PROOF.
C, so there exists a projective group subalgebra
for
UIu
UIU
such that
be a Galois extension over
generated by
ble,
Eext, let @’, /’
Then
A.
c; and so
Thus cz
in
for all
in
from
/
151
utuj
:t
U.
o
CGf)o
is in
r
for
By hypothesis,
center
is
Thus the center of
C, so r is in C.
C.
Clearly, C is contained in the center of CGf, so
has center C.
is an Azumaya algebra over C.
Therefore,
has
CGf
contained in
CGf
CGf
But then
restricted to
G’
to
by Theorem 3 in [I]
CGf
AG’ Uc
restricted to
Lemma 3 I,
G’
A U
AG’u
A G’
is Galois over
Moreover,
G’
let
{xi, Yi
in
CG
with Galois group
C
Since
is isomorphic with
(for the Galois set for
AG’u)
_
is a central Galcis algebra over
CGf
CGf
G’
G’
-A U
restricted to
with Galois group
G’
6’
such that
CGf
by
restricted
is also a Galois set for
fG’ U
A
/
,k}
i
be a Galois
By hypothesis, A is Galois over A G’ with
U
Galois group G’
so G’
G’ restricted to
Thus
A G’ U by Lemma 3.2
is also a Galois set for A over A
Therefore, A is finitely generated and projective over A G’ with a dual basis
(tr)Yi__}
where tr
(the trace of G) ([2], see the proof of Theorem I, P. 119)
G’
Hence, for any a in A, a
Thus
xitr(Yia) which is in
U
G’
A
U.
We recall that a Galois extension T over T G with Galois group G
is called a centralized Galois extension if A G has the same center as A.
COROLLARY 3.4.
By keeping the hypotheses of Theorem 3.3, A is a centraset for
@A
over
A
G’
G’.
xi’ Yi}
xi,
A
A
lized Galois extension.
PROOF.
for any
r
Theorem 3 3,
also C.
Clearly,
C
is contained in the center of
A G’
in the center of
A
G’
#A U,
so
r
r
is in
C.
G’.
A
G’
Next is a condition under which A is a projective group ring.
By keeping the hypotheses and notations of Theorem 3 .9 if
G’
G’
such that A
is an AzuU is separable over C, then A A G’
THEOREM 3 5
A
A
Conversely,
U. By
G’
is
Thus the center of A
is in the center of
Gf
152
G. SZETO AND L. MA
maya C-algebra.
y
PROOF
bra
ing an Azumaya subalgebra
CGf,
G’
A
so
containing an Azumaya alge-
U
is an Azumaya C-algebra contain-
A
By hypothesis,
C.
over
CGf
A
A
3.3,
heorem
ZA(CGf)cCG f
ZA(CG f) is
where
ZA(CG f)
is
C-subal6.eCGf
4.3, Pc 57).
Theorem
algebras
Azumaya
(3,
commutant
for
theorem
bra by the
the commutant of
in
A
such that
A G’ we conclude that
Noting that
G
is an Azumaya’C-algebra.
that A
A G’
A
ZA(CGf)
an Azumaya
A
CGf
G’
such
Gf
The following is a property of a Galois projective group ring.
(not necessarily commutative).
Let R be a ring with
THEOREM 3.6.
G’
G’
then
with Galois group G’
is Galois over (RGf)
If
(RGf)
(RGf) G’.
RGf
Since
C C
Hence
are free over C by the proof of Lemma 3.2.
is Galois,
Next, let D
of
there exists a projective group subalgebra
In fact, for any x
D.
We claim that C
be the center of R.
Let
PROOF.
be the center of
C
Then
RGf.
{Ua}
RGf
RGf.
CGf
in
C
C
t
and
in
DGf.
xt
R,
ix, so r
Since
rU
But then
are in the center of R.
are free over C and D res-
U}
Hence
D.
Thus the center of
Clearly, D C, so C
pectively, C C D.
is C.
Noting that the order of G is invertible we conclude that CG f
is a central Galois C-algebra with
is an Azumaya C-algebra; and so
such
Moreover, since
Galois group G’
([13, Theorem 3).
f
d
in
an
element
exists
there
that C is a C-direct summand of CGf,
CGf DGf.
CGf
CGf
RcCG
RGf
HOmAG,(A,AG’)
by using the fact that
tr(d)
such that
[2], P. 119
@’O@
Then, for any
and the introduction to Section 2 in [51).
G’
and
in
in G,
(RGf)
rE(U)tr(d)
r(R)tr(Uwd)
Thus (RGf) G’
R.
CGf
(see
(tr)A
(r(R)U)= (U)
R (for tr(Ud)
which is in
is in
C).
We now generalize the theorem of F.R. DeMeyer that if KGf is Azumaya
over a commutative ring K, then KGf is Galois over K with Galois group
G’
(I],
THECF24
Theorem 3).
3.7.
AG’u G
If
is an Azumaya C-algebra with
G’
U}
a free set
and is a centralized Galois extension over
@A U
over C, then
A Gf
A G’ with Galois group G’.
G’
contains a projecare free over C
Since
PROOF.
tive group algebra
of
G is invertible, CGf is
CGf. Since the order
is in the center of
separable over C.
Since the center of
is a
it is equal to C.
Thus
is an Azumaya C-algebra.
Hence
central Galois C-algebra with Galois group G’
(g1, Theorem 3, and Lemma
G’
we conclude that
Noting that
3 I)
U@ is Galois
G’
G’
over A
with Galois group G’.
Moreover, since
Uo is Azumaya
G
G
theorem for Azucommutant
the
over C by hypothesis,
by
U A
maya algebras as given in the proof of Theorem 3.5.
Thus the proof is corn-
U
A U
AG’u,
CGf
CGf
CGf
CGf_,AG’u,_.
A
A
A
Gf
plete.
COROLLARY 3.8.
Let
R
be a ring with
and
C
the center of the pro-
GALOIS PROJECTIVE GROUP RINGS
153
If RGf is an Azumaya algebra such
E.
C, then RGf is a centralized Galois extension over R with Galois group G’.
Then there exists a projecLet the center of R be D.
PROOF.
Since
tive group algebra DGf in RG.
Clearly, RGf
D by a similar
are free over C by hypothesis, we can show that C
for each a in R
proof of Theorem 3.6.
Moreover, since
jective group ring
U
that
over
RGf
are free over
RDDG.
aU
C in
Ua
G, the center of DGf is contained in C.
he center of
contained in the center of DGf, so C
and
the center of
fore
is
RGf
DGf.
Thus
is Galois over
R
DGf
Clearly,
DGf.
C
Hence
is a central Galois D-algebra.
by Theorem
3.6
such that the center of
is
D
There-
R
C.
We conclude the paper with two more properties of a Ga!ois projec-
rive group ring
RGf.
Let RGf be a Galois projective group ring with Galois
THEOREM 3.9.
the centraliThen (I)
group G’ over a ring R and with center C.
zer of the projective group algebra CGf in RGf is R, and (2) the
center of R is equal to C (and hence RGf is a centralized Galois
extension over R).
G’
is an inner
Since G’
PROOF.
(I) By Theorem 3.6, R
(RGf)
induced by U), part (I) is immediate.
automorphism group of
RGf
Then it is easy to verify that DG f
(2)
Let D be the center of R.
the centralizer of R in
RGf. By part (I), DGf CGf. Since RGf
But then
is Galois,
are free over C by the proof of Lemma 3.2.
U
.D
C.
REFERENCES
I.
2.
3.
4.
Galois Theory in Separable Algebras over Commutative
Rings, Illinois J. Math. 10 (1966), 287-295.
DeMeyer, F.R. Some Notes on the General Galois Theory of Rings,
Osaka J. Math. 2 (1965), 117-127.
DeMeyer, F.R. and E. Ingraham. Seoarable Algebras over Commutative
Rin__s, Springer-Verlag, Heidelberg-New York, 1971.
Chase, S., D. Harrison and A. Rosenberg. Galois Theory and Galois
Cohomology of Commutative Rings, Mem. Amer. Math. Soc. No. 52
DeMeyer, F.R.
(1965).
5.
6.
7.
Miyashita, Y. Finite Outer Galois Theory of Non-commutative Rings,
J. Fac. Sci. Hokkaido Univ. Set. I, 19 (1966), 114-134.
Szeto, G. A Characterization of a Cyclic Galois Extension of Commutative Rins, .J. Pur..e ..and Applied Algebra 16 (1980), 315-322.
Szeto, Go On Separable Abelian Extensions of Rings, Internat. J.
Math. and Math. Sci. 4 (1982), 779-784.