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The Cartan-Brauer-Hua Theorem

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Jan Treur at Vrije Universiteit Amsterdam
Jan Treur
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This paper presents a new proof for the Cartan-Brauer-Hua Theorem.
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MATHEMATICS
THE CARTAN-BRAUER-HUA THEOREM
BY
JAN TREUR
(Communicated by Prof. H. Freudenthal at the meeting of January 29, 1977)
In this note we present an elementary proof which hardly uses any
computation; cf. [a] p. 2. First a lemma.
LEMMA:
Suppose K is a skew field and E is a left K-linear space.
Assume M C E is closed under addition and contains at least two inde-
pendent elements over K. Let rj: M -+ E be
a
map satifying $(a+ b) =
= d(a) +4(b) and &a) = 31cc a for all a, b E M and certain ila E K. Then
there exists one element 1 E K such that $~(a)=Ja for all a E M.
PROOF:
Let a,
b
E M be nonzero ; choose c E M such that a and c are
independent. From &a + A& =+(a) + 4(c) = +(a + c) = jla+Ja + c) it follows
that A,=1 a+c=Jc. Since
b
is independent of a or c, also jib = 3La = 1,.
THEOREM
(Cartan-Brauer-Hua) : Suppose L/K is an extension of skew
fields and M C L is closed under addition and 1 E M. If aK C Ka for all
a E M then either M C K or M C ZL(K) (h ere ZL(K) denotes the centralizer
of K in L). In particular, if aKa-l= K for all nonzero u E L then either
K = L or K is contained in the center of L.
PROOF:
Suppose M is not contained in K, then M contains at least
two elements left independent over K. Let
b E
K be given; for each
a
E M,
ab E aK C Ka. Hence ab=&a for some 1, E K. Apply the lemma to the
right multiplication rb : a I+- ab : M -+ L. We have an element il E K such
that ab=la for all a E M. In particular this holds for a=
1,
hence 2= b.
Therefore ab= ba for all a E M and b E K. For the last statement, take
M=L.
REMARK:
The above proof remains valid if L is replaced by any ring
extension of the skew field K.
Mathematkch Inetituut
De Uithof, Utrecht
454
REFERENCES
1. Cartan, H. - Theorie de Galois pour les corps non commutatifs, Ann. Sci. Ecole
Norm. Sup. 64, 59-77 (1947).
2. Brauer, R. - On a theorem of H. Cartan, Bull. Amer. Math. Sot. 55, 619-620
(1949).
3. Hua, L. - Some properties of a sfield, Proc. Nat. Acad. Sci. USA 35, 533-537
(1949).
4. Treur, J. - A duality for skew field extensions, Dissertation, Utrecht, 1976.
... Schenkman and Scott [11] proved that if R is a subdivision ring of D such that U (R) is a subnormal subgroup of U (D), then either R = D or R is in the center of D. Wähling [13] proved that if D is a unitary ring without zero divisors and R is a division subring of D which is D-invariant, then either R is in the center of D or R = D. For more generalizations see [9,12]. Also we need to recall the following theorem of Asano [2], which has a similar idea. ...
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