Finitely generated special Jordan and alternative $PI$-algebras

The author explores the question of whether identities related to special Jordan and alternative $PI$-algebras exist in associative algebras. It is proved that if $A$ is a finitely generated special Jordan (alternative) $PI$-algebra, then the universal associative enveloping algebra $S(A)$ (respectively, the universal algebra $\mathscr R(A)$ for right alternative representations) of algebra $A$ is also a $PI$-algebra. As a corollary it is proved that the upper nilradical of a finitely generated special Jordan or alternative $PI$-algebra over a Noetherian ring is nilpotent. A similar result holds for the Zhevlakov radical of a finitely generated free alternative algebra. In addition, a criterion is obtained for local associator nilpotence of an alternative algebra.
Bibliography: 19 titles.