# A radical approach to algebra by Mary W Gray By Mary W Gray

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Example text

We define the trace tr : A0 -> F by putting tr d = Tr d , the ordinary trace of the matrix d E Do c Mn (F) , and tr r = 0 if r E A0 n Rad A. It is well §3. TRACE IDENTITIES 53 known that the algebra Mn(F) satisfies the Cayley-Hamilton identity with trace xn (x)x = 0, where xn (x) is the Cayley-Hamilton polynomial (with trace) defined recursively by the formulas xo(x) = 1 and xn('x) = xn-1(X) X - n (28) JC). It follows from this and the definition of the trace on AO that xn(a)a E AO n Rad A for any a E AO .

S. algebra A . We call the set com(IF, d(F (t(IF, the relative complexity of IF over r. s. algebra A. We define a partial order on the set of types, by setting (al, a2, a3) < if and only if a! < /3t for i = 1, 2, 3. We compare the (/31, /3z, /33) complexities in the following manner: (a1, a2 , a3 , a4) < (91 QZ /33' 9a) if and only if either (al, a2, a3) < A 182' 93) or (al, a2, a3) = (/31, /3z, 93) and a4 < /34. 2 that if I'(') D 1 I'(2) D TZ[A] 2[A], r(r(`) < r(r(2' t(I'(1), I'(Z)) < t(I'(2,) < t(A), com(I'(I) , I"(Z,) < com(I'(Z)) < com(A) com(I'(l)) , < com(I'(Z)).

Ykak(n+1)-1 0 (1) where y, E Y and xj E X, is satisfied by A. s. algebra, we have A = D + Rad A, where D is a semisimple graded subalgebra. It follows from the definition of the parameter a(A) that Do C Mn (F) , (n = a(A)). Therefore each element of Do is algebraic over F of degree not larger than n . Then, since the elements d'1, d ' ' , ... , d , 1 are linearly dependent in A" , for any a, , ... , an E A x the identity (_1)adt7(1)-la1dQ(2)-la2 E QES(n+l) ... an dQ(n+l)-1 E (_1)aya(l)alya(2)a2 ... 