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Products of several commutators in a Lie nilpotent associative algebra. (English) Zbl 1379.16014

Summary: Let \(F\) be a field of characteristic \(\neq 2,3\) and let \(A\) be a unital associative \(F\)-algebra. Define a left-normed commutator \([a_1,a_2,\dots,a_n]\) \((a_i\in A)\) recursively by \([a_1,a_2]=a_1a_2-a_2a_1\), \([a_1,\dots,a_{n-1},a_n]=[[a_1,\dots,a_{n-1}],a_n]\) \((n\geq 3)\). For \(n\geq 2\), let \(T^{(n)}(A)\) be the two-sided ideal in \(A\) generated by all commutators \([a_1,a_2,\dots,a_n]\) \((a_i\in A)\). Define \(T^{(1)}(A)=A\).
Let \(k,\ell\) be integers such that \(k>0\), \(0\leq\ell\leq k\). Let \(m_1,\dots,m_k\) be positive integers such that \(\ell\) of them are odd and \(k-\ell\) of them are even. Let \(N_{k,\ell}=\sum_{i=1}^km_i-2k+\ell+2\). The aim of the present note is to show that, for any positive integers \(m_1, \dots,m_k\), in general, \(T^{(m_1)}(A)\dots T^{(m_k)}(A)\nsubseteq T^{(1+N_{k,\ell})}(A)\). It is known that if \(\ell<k\) (that is, if at least one of \(m_i\) is even), then \(T^{(m_1)}(A)\dots T^{(m_k)}(A)\subseteq T^{(N_{k,\ell})}(A)\) for each \(A\) so our result cannot be improved if \(\ell<k\).
Let \(N_k=\sum_{i=1}^km_i-k+1\). Recently, Dangovski has proved that if \(m_1,\dots,m_k\) are any positive integers then, in general, \(T^{(m_1)}(A)\dots T^{(m_k)}(A)\nsubseteq T^{(1+N_k)}(A)\). Since \(N_{k,\ell}=N_k-(k-\ell-1)\), Dangovski’s result is stronger than ours if \(\ell=k\) and is weaker than ours if \(\ell\leq k-2\); if \(\ell=k-1\), then \(N_k=N_{k,k-1}\) so both results coincide. It is known that if \(\ell=k\) (that is, if all \(m_i\) are odd) then, for each \(A\), \(T^{(m_1)}(A)\dots T^{(m_k)}(A)\subseteq T^{(N_k)}(A)\) so in this case Dangovski’s result cannot be improved.

MSC:

16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16R40 Identities other than those of matrices over commutative rings
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