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Central simple algebras with involution viewed through centralizers.pdf
JOURNAL OF ALGEBRA 63, 41-55 (1980)
Central Simple Algebras with
Involution Viewed through Centralizers
LOUIS HALLE ROWEN*
Bar Ilan University, Ramat Gan, Israel
Communicated by I. N. Herstein
Received April 24, 1979
TO NATHAN JACOBSON ON HIS 70TH BIRTHDAY
This paper is motivated by the use of rings with involution of the second kind
in the theory of central simple algebras of exponent 2; viz., Albert [l], Racine
[ll], Tignol [16], and Amitsur et al. [5, Sect. 61. We present the idea after
recalling some basic facts. An involution is an antiautomorphism of degree 2.
If R is simple finite dimensional over its center F (written [R] E Br(F)) then any
involution induces an automorphism of F of degree 2; we say the involution
has $rst (resp. second) kind if this induced automorphism has degree 1
(resp. 2).
R has an involution of the first kind iff R OF R is a matrix algebra, i.e., [R]
has exponent 2 in l(F). Write lb,(F) for (all [R] E h(F) having exponent 2)
Also, if (*) is an involution of R, write Z(R, *) = {o E F ] 01* = o]; conversely
if [F: C] = 2 write Br,(F/C) = ([RI E B,(F) 1R has an involut
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