: C(K;D) D . is a left D, > module: D, KeD but also the subfield K(a) S C(K) is contained Then K(a) c D is a commutative subfield of D <~: If is a maximal subfield < a-I E C(K). properly c E K. t : D -> be any division ring and Then K[a] S C(K), s ::;Lb @ c E D ~ K, As a consequence of the commutativity of D @ K D . F, and take any element K. Let K. ~ K, which is a sum of a finite number of terms D, k LEMMA.

CROSSED subfield ring bO*T right baT should D KeD with center such that F which F c K will be continued. different operators: oTb = baT = = o(bT) (bo)T = = = O,T PRODUCTS Any division contains is a finite, group operations (G,'") be regarded a normal -1 S S = and k -l S S = 1, and G (bT)o (bo)T G; b € (K/F,o,g) NOTATION. Throu~hout, of arbitrary characteristic. l F wilt be a fixed field All algebras £ center A ; A will be over as before \ F [A:F] denotes € K. > K be any automorphismwith inverse ...

Of the structure and properties Let PRELIMINARIES. with unity F'l £ A = and 1 elements F = F'l E <: A of division B and A and 1 <: be any algebras B so that All tensor products B. rings D. 1. . c D e K over F 1. 0 s k s m-l. co(m+l)co(m)N(C). Since as a K-vector Irrespective K or even F, such D, it will be shown that the tensor product that o(i+j-l) - C » = m+k - D. D Then central role in determining the struc- may be regarded as a dense ring of c(o(i»o(j)c(o(;» c(o(i+j» center F. 1. MAXIMAL SUBFIELDS III-3.