Bernstein algebras that are algebraic and the Kurosh problem

We study the class of Bernstein algebras that are algebraic, in sense each element generates a finite-dimensional subalgebra. Every algebra has maximal algebraic ideal, and quotient is zero-multiplication algebra. Several equivalent conditions for to be given. In particular, known characterizations train terms nilpotency generalized case locally algebras. Along way, we show if Banach (respectively, train), then it bounded degree train). Then investigate Kurosh problem algebras: whether finitely generated which finite-dimensional. This turns out have closed link with question about associative when bar-ideal nil, asks prove answer positive some specific cases low degrees, construct counter-examples general case. On other hand, by results Yagzhev Jacobian conjecture certain statement Engel nilpotence identities multioperator quadratic mappings holds