𝑃𝐼-algebras satisfying identities of degree 3

Klein, Abraham A.

doi:10.1090/S0002-9947-1975-0349741-5

$\textrm {PI}$-algebras satisfying identities of degree $3$
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by Abraham A. Klein PDF

Trans. Amer. Math. Soc. 201 (1975), 263-277 Request permission

Abstract:

A method of classification of PI-algebras over fields of characteristic 0 is described and applied to algebras satisfying polynomial identities of degree 3. Two algebras satisfying the same identities of degree 3 are considered in the same class. For the degree 3 all the possible classes are obtained. In each case the identities of degree 4 that can be deduced from those of degree 3 have been obtained by means of a computer. These computations have made it possible to obtain-except for three cases-all the identities of higher degrees. It turns out that except for a finite number of cases an algebra satisfying an identity of degree 3 is either nilpotent of order 4, or commutative of order 4, namely the product of 4 elements of the algebra is a symmetric function of its factors.

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