Abstract
ABSTRACTLet K be a unital associative and commutative ring and let K⟨X⟩ be the free unital associative K-algebra on a non-empty set X of free generators. Define a left-normed commutator inductively by , . For n≥2, let T(n) be the two-sided ideal in K⟨X⟩ generated by all commutators .It can be easily seen that the ideal T(2) is generated (as a two-sided ideal in K⟨X⟩) by the commutators . It is well known that T(3) is generated by the polynomials and . A similar generating set for T(4) contains 3 types of polynomials in xi∈X if and 5 types if . In the present article, we exhibit a generating set for T(5) that contains 8 types of polynomials in xi∈X.
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