Let
F be a field and let
8b4bb1400b8804c2576c0a65323de" title="Click to view the MathML source">F〈X〉 be the free unital associative algebra over
F freely generated by an infinite countable set
b5b9b5da02245a4a101237efcb7" title="Click to view the MathML source">X={x1,x2,…}. Define a left-normed commutator
[a1,a2,…,an] recursively by
e688a19ca5" title="Click to view the MathML source">[a1,a2]=a1a2−a2a1,
[a1,…,an−1,an]=[[a1,…,an−1],an] (
e74d64ee0f921b3c0832fe2f49c" title="Click to view the MathML source">n≥3). For
8e0e1c6b5bd1b4715c9b15070baf0" title="Click to view the MathML source">n≥2, let
8b09368b699d4c4a7f09fbd1f874a" title="Click to view the MathML source">T(n) be the two-sided ideal in
8b4bb1400b8804c2576c0a65323de" title="Click to view the MathML source">F〈X〉 generated by all commutators
[a1,a2,…,an] (
980dd448862d0519211a0439eb9be" title="Click to view the MathML source">ai∈F〈X〉).
Let F be a field of characteristic 0. In 2008 Etingof, Kim and Ma conjectured that T(m)T(n)⊂T(m+n−1) if and only if m or n is odd. In 2010 Bapat and Jordan confirmed the “if” direction of the conjecture: if at least one of the numbers m, n is odd then T(m)T(n)⊂T(m+n−1). The aim of the present note is to confirm the “only if” direction of the conjecture. We prove that if 8bb7935aae41b032c154e" title="Click to view the MathML source">m=2m′ and e7d10791413a383d493235d" title="Click to view the MathML source">n=2n′ are even then e713fdd9f3f456" title="Click to view the MathML source">T(m)T(n)⊈T(m+n−1). Our result is valid over any field F.