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Abstract. Let Z X be the free unitary associative ring freely generated by an infinite countable set X = {x1, x2, . . .(n) be the two-sided ideal in Z X generated by all commutators [a1, a2, . . . , an] (ai ∈ Z X ). It can be easily seen that the additive group of the quotient ring Z X /T (2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have noted that the additive group of Z X /T (3) is also free abelian. In the present note we show that this is not the case for Z X /T (4) . More precisely, let T (3,2) be the ideal in Z X generated by T (4) together with all elements [a1, a2, a3][a4, a5] (ai ∈ Z X ). We prove that T (3,2) /T (4) is a non-trivial elementary abelian 3-group and the additive group of Z X /T (3,2) is free abelian.

Available online xxxx Communicated by Michel Van den Bergh MSC: 16R10 16R40 Keywords: Polynomial identities T -ideals Torsion elements Lie nilpotent ringLet Z X be the free unital associative ring freely generated by an infinite countable set X = {x 1 , x 2 , . . .}. Define a leftnormed commutator [a 1 , a 2 , . . . , a n ] inductively by [a, b] = ab − ba, [a 1 , a 2 , . . . , a n ] = [[a 1 , . . . , a n−1 ], a n ] (n ≥ 3). For n ≥ 2, let T (n) be the two-sided ideal in Z X generated by all commutators [a 1 , a 2 , . . . , a n ] (a i ∈ Z X ). Let T (3,2) be the two-sided ideal of the ring Z X generated by all elements [a 1 , a 2 , a 3 , a 4 ] and [a 1 , a 2 ][a 3 , a 4 , a 5 ] (a i ∈ Z X ). It has been recently proved in [22] that the additive group of Z X /T (4) is a direct sum A ⊕ B where A is a free abelian group isomorphic to the additive group of Z X /T (3,2) and B = T (3,2) /T (4) is an elementary abelian 3-group. A basis of the free abelian summand A was described explicitly in [22]. The aim of the present article is to find a basis of the elementary abelian 3-group B.

Let $F$ be a field and let $F \langle X \rangle$ be the free unital associative algebra over $F$ freely generated by an infinite countable set $X = \{x_1, x_2, \dots \}$. Define a left-normed commutator $[a_1, a_2, \dots, a_n]$ recursively by $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \dots, a_{n-1}, a_n] = [[a_1, \dots, a_{n-1}], a_n]$ $(n \ge 3)$. For $n \ge 2$, let $T^{(n)}$ be the two-sided ideal in $F \langle X \rangle$ generated by all commutators $[a_1, a_2, \dots, a_n]$ ($a_i \in F \langle X \rangle)$. Let $F$ be a field of characteristic $0$. In 2008 Etingof, Kim and Ma conjectured that $T^{(m)} T^{(n)} \subset T^{(m+n -1)}$ 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 $T^{(m)} T^{(n)} \subset T^{(m + n -1)}.$ The aim of the present note is to confirm the "only if" direction of the conjecture. We prove that if $m = 2 m'$ and $n = 2 n'$ are even then $T^{(m)} T^{(n)} \nsubseteq T^{(m +n -1)}.$ Our result is valid over any field $F$.Comment: 8 pages, remarks and references added, typos fixe

Let F be a free group and let w A F . For a group G, let G w denote the set of all w-values in G and wðGÞ the verbal subgroup of G corresponding to w. A word w is called boundedly concise if, for each group G such that jG w j c m, we have jwðGÞj c c for some integer c ¼ cðmÞ depending only on m. The main theorem of the paper says that if w is a boundedly concise word and G is a group such that jx Gw j c m for all x A G then jx wðGÞ j c d for all x A G and some integer d ¼ dðm; wÞ depending only on m and w.