# PI-Algebras: An introduction by Prof. Nathan Jacobson (auth.)

By Prof. Nathan Jacobson (auth.)

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I) n det a. Consider the polynomial ring Let g(~l' "'''~n ) ~[~i' "''~n]" be a symmetric polynomial contained in =DI"'Dn symmetric polynomials g and Pl'''''Pn h where is unique since the elementary are algebraically independent. ,deta) Mn(K ) into K. In particular, we can take d = ~ [ (7i - ~ j ) 2 i

For any positive , U2i Bn-iAB i , i _< i _< n. = we have UIU 2 .. U h = (Bn-lA)hB[h/2] and for any j > k (22) Let ideals of of define h < 2n (21) B B UjU k C A B n nA. be chosen to be the smallest positive integer such that is nilpotent, Since B is nilpotent such an n satisfies a strongly regular identity of degree is multilinear and on dividing by a unit in identity f (23) n > [d/2] above formulas, (24) d Since A we may assume this we may assume the has the form XlX 2 ... x d - Now suppose K exists.

The proof. ,am) ai -~Aa on the right, on the degree we may assume it follows that ai -~Aa This shows that for any is the right annihilator of where Aa/Z Thus satisfies Using induction is locally nilpotent. Since This completes 35 THEOREM 2. Bet identity of degree B [d/2] ~ N ( O ) A d. be an algebra satisfying a strongly re~ul9 ~ Then any nil subalgebr a the sum of the nilpotent Proof. Suppose first that integer n (20) U2i_l -_ Bn-iABi-i For A satisfies A. is nilpotent. For any positive , U2i Bn-iAB i , i _< i _< n.