Renumerating elements we let [gkl''''gkm} be all non-conjugate ele- ments of each other; then {Ck. ,...,C k } are all different elements in m m the set, {CI,...,Cn}.
I nternat. J. Math. & th. Sci. (1978)433-438 Vol.
433
ON SEPARABLE EXTENSIONS OF GROUP RINGS AND QUATERNION RINGS
GEORGE SZETO Mathematics Department Bradley University Peoria, Illinois 61625 U.S.A.
(Received November 29, 1977)
ABSTRACT.
lqe
purposes of the present paper are (i) to give a necessary and
sufficient condition for the uniqueness of the separable idempotent for a
separable group ring extension RG (R may be a non-commutative ring), and
(2) to give a full description of the set of separable idempotents for a quaternion ring extension RQ over a ring
R, where Q are the usual quaternions i,j,k
and multiplication and addition are defined as quaternion algebras over a field.
We shall show that RG has a unique separable idempotent if and only if G is abelian, that there are more than one separable idempotents for a separable quaternion ring RQ, and that RQ is separable if and only if 2 is invertible in
No
KEY WORDS AND PHRASES.
Group Rings, Idempotents in Rings, Separable Algebras
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
Major:
16A26
Minor:
16A16, 16A32
Go SZETO
4 34
INTRODUCTION. M. Auslander and O. Goldman bras over a commutative ring.
([I]
and
[2])
studied separable alge-
Subsequently, the investigation of se-
parable algebras (in particular, Brauer groups and Azumaya algebras)
has attracted a lot of researchers, and rich results have been obtained
(see References).
K. Hirata and K. Sugano
([5])
generalized the con-
cept of separable algebras to separable ring extensions; that is, let S be a subring of a ring T with the same identity.
Then T is called a
separable ring extension of S if there exists an element
TST
aib i
in
x([aib i) ([aibi)x for each x in T and [aib i element aib i is called a separable idempotent for T. We
I.
such that
Such an
note that a separable idempotent takes an important role in many theoreins
(for example, see [6], Section 5,6, and 7).
that
(I/n)(gig1)
and
ei1eli
([4],
It is easy to verify
Examples II and
III, P. 41)
are separable idempotents for a group algebra RG and a matrix ring M
respectively, where G
{ gl,...,g n}
(R)
with n invertible in R and e.. are
m3
We also note that the separable idempotent for a commu-
matrix units.
tative separable algebra is unique
2.
m
([6],
Section
I, P. 722).
PRELIMINARIES. Throughout, G is a group of order n, R is a ring with an identity
I.
The group ring RG
R-module with a basis
ris j
rig i / r i in R and gi in G}, which is a {gi and (rig i)(sig i) tkg k where t k
for all possible i,j such that
ded in RG by
r-rgl,
tiplication map
where
RGRRG--RG
gl
gig j
.
gk"
The ring R is imbed-
is the identity of G
is denoted by
free
(gl
Clearly,
I).
{gigj
The mul-
/ i,j
SEPARABLE EXTENSIONS OF RINGS
,... ,n
RGRRG.
form a basis for
is called a commutant element in
(rij(gigj))x
An element
RGRRG
if
435
RGRRG
in
rij
x(rij (gigj))
for all x in RG.
MAIN THEOREMS. We begin with a representation for (x) for a commutant element x in
RGRRG
and then we show that RG has a unique separable idempotent
if and only if G is abelian.
LEMMA I.
RGRRG.
ment in
i=ml
(x)
Then
of conjugate classes of G, and C
k
nk.
rlk:I )nk.i Ck.i
where m is the number
is the order of the normalizer of
--.,
is the sum of different conjugate elements of
gk.’
for some k.
i
{1,...,n.
1
PROOF.
Since x is a commutant element,
the same term in
gk"
gqgp
Thus
gk;
gpX
xg
P
for each
gp
in G.
gpx is rlk, and the coefficient of x gp is rpq, where gqgp gk" Hence rlk rpq whenever x krlk (gpgq), where p,q run over 1,...,n, such
The coefficient of the term
that
1,...,n, be a commutant e!e-
i,j
1
and k: in
gqgp
.
ri.i (gi g.i )’
Let x
gpgk
krlk(pgpgkgp-I).
that is, x
-I).
in
For a fixed k
-I
pgpgkgp
TakingS(x)
nkCk
where n
k
is the
order of the normalizer of S and C is the sum of all different conjuk k
gate elements of gk" es form a partition of
Renumerating
elements
Hence ,(x)
G, C i we let
ments of each other; then the
set,
{CI,...,Cn}.
[gkl’’’’gkm }
{C k ,...,C k } m
Ck.,
Since conjugate class-
C. if and only if gi is conjugate to
Thus (x)
coefficients of the same
k=]rknkCk.
m
gj.
be all non-conjugate ele-
are all different elements in
i:i (rlk :)nk.Ck.
where
rlk
are
and m is the number of conjugate classes
4 36
G. SZETO
of G.
THEOREM 2.
Then, RG has a
Let RG be a separable extension of R.
unique separable idempotent if and only if @ is abelian.
Let x
PROOF.
Then by the lemma,
Zrij(gig j) be a separable idempotent m (x) i:I (rlk!I )nk. Ck.! where C k. is i
all conjugate elements of
Ckl and that
rlln
and n
k
.
we have
I.
(rlk,)n k i i
rlk
n,
n,
.
(rlk,)nk.C,. I 1
(x)
C
0 for each i
k
6
1,
R.
Toting
Thus
nk.,
bemng
But conjugate classes form a
R.
rlk
0 implies that
I
rlk
0 for each i
0 in the unknowns 1
has trivial solutions if and only if n
m, and this
Since the uniqueness of the separ-
holds if and only if G is abelian. able idempotent (=
1.
i
This system of homogeneous equations
with i
Then
(rlkl)nkl Ckl
Since
is invertible in
is also invertible in
G, so
the identity of G.
and so the first equation becomes
r11
the order of G,
partition of
rlk
and
the sum o
i
I,
Let gk
the order of G.
(rlk)n kl I, Ckl Hence
a factor of
i
n,
kl
g.. m
for RO.
(I/n)(gig1))
is equivalent to the existence of
trivial solutions of the above system of equations, the same fact is equivalent to G being abelian.
The theorem tells us that there are many separable idempotents for a separable group ring RG when G is non-abelian. if RG is a separable extension of
from the proof of the theorem.
R,
the order of G is invertible in R
Next, we
discuss another popular separ-
able ring extension, a quaternion ring extension
(r1+rii+rjj+rkk
/ i,j,
Also, we remark that
and k are usual
RQ, where RQ
quaternions).
(RQ,+.)
is a ring
extension of R under the usual addition and multiplication similar to quaternion algebras over a field.
Now we characterize a separable idem-
437
SEPARABLE EXTENSIONS OF RINGS
potent for a separable quaternion ring extension RQ.
Let RQ be a separable quaternion ring extension.
THEOREM 3.
a commutant element x
rst(St),
s,t
able idempotent for RQ if and only if
1,i,j,k, in
efficients of the term 191 on both sides are
so
r
ril
rjk.
-ril
Hence
by comparing coefficients of other
-rkk rlj
rjl
rpq
if ts
x
r
-rki
rik
and
rli
rlk
rkl
have
rli. -ril
-rjj
-rii
In other words,
rji.
-ij+ ji Ik 1k+k
*
r114+rliO+rljO+rlkO 4r11. idempotent if and only if r11 I/4
PROOF.
Consequently, x is a se-
3
r11
I/4, rli
The sufficien-
rlk
rlj
is a separable idempotent for
0 as
RQ.
It is easy to see that every x of the form (*) in Theorem
r11 rli rlj and rlk in the center of R RQRR Q. Hence, from the proof of Theorem 3,
with
mutant elements isif qp
Then RQ
R.
The necessity is immediate from the theorem.
given in (*) in Theorem
REMARK.
I).
(for (x)
Let RQ be a quaternion ring extension of R.
cy is clear since the element x with
Pq
ril
Similarly,
-rjk.
r11
-rij
is separable if and only ,if 2 is invertible in
-r
so
The co-
qp, and
COROLLARY 4.
in
-rli
rkj
ril
terms, we
But then (x)
3
and
xi.
rst -rpq if ts -qp. Thus (11-ii-jj-kk)+rli( 1i+i1-jk+kj)+rlj(1j+j1-ki+ik)+
r
parable
-ril
ix
xk, so the coefficients of the term j1
Also, kx
k0
on both sides are
rst
RQRQ,
xj, the coefficients of the term k1 on both sides are
Since jx
-rkj
is a separ-
I/4.
r11
Since x is a commutant element in
PROOF.
RQRR Q
Then
-ts}.
C
(rst(St)
/ r st
r
pq
is a commutant element
the complete set of cornif qp
ts, and r st
Also, the complete set of separable idempotents for
G. SZETO
4 38
RQ is a subset of C such that r 11 I/4 and r Ii’ rlj’ rlk are in the center of R. Thus there are many separable idempotents.
REFERENCES
I.
Auslander, M. and O. Goldman. The Brauer Group of a Commutative Ring, Trans. Amer. Math. Soc. 97 (1960) 367-409.
2.
Auslander, M. and O. Goldman.
Maximal Orders, Trans.
Amer. Math.
So__c. 97 (1960) 1-24. Lectures on Topics in Algebraic K-Theory, Tata Institute of. Fundamental Research, Bombay, 1967.
3.
Bass, H.
4.
DeMeyer, F. and E. Ingraham.
5.
Hirata, K. and K. Sugano. On Semisimple Extensions and Separable Extensions over Non-Commutative Rings, J. Math. Soc. Japan-18
Separable Algebras Over Commutative Rings, Springer-Verlag, Berlin-Heidelberg-New York, 181, 1971.
(1966) 360-373.
6.
Villamayor, O. and D. Zelinsky. Galois Theory for Rings with Finitely Many Idempotents, Nagoya Math. J. 27 (1966) 721-731.
7.
Zelinsky, D. Brauer Groups, Springer-Verlag, Berlin-HeidelbergNew York, 549, 1976.
