RELATIVELY FREE ASSOCIATIVE LIE NILPOTENT ALGEBRAS OF RANK 3

Let Phi be an arbitrary unital associative and commutative ring. The relatively free Lie nilpotent algebras with three generators over Phi are studied. The product theorem is proved: (TT(m))-T-(n) subset of T(n+m-1) where T-(n) is a verbal ideal generated by the commutators of degree n. The identities of three variables that are satisfied in a free associative Lie nilpotent algebra of degree n >= 3 are described. It is proved that the additive structure of the considered algebra is a free module over the ring Phi.