  We introduce an embedding of the free magma on a set A into the direct
product of the free magma on a singleton set and the free semigroup on A. This
embedding is then used to prove several theorems related to algebraic
independence of subsets of the free non-associative algebra on A. Among these
theorems is a generalization of a result due to Kurosh, which states that every
subalgebra of a free non-associative algebra is also free.

Rings and Algebras