Abstract:
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: $T^{(n)}T^{(m)} \subseteq T^{(n + m-1)}$,
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 $n\geq 3$ are described.
It is proved that the additive structure of the considered algebra is a free module over the ring $\Phi$.
Keywords:associative Lie nilpotent algebra, identity in three variables, torsion of a free ring.
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