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 field and let $F \langle X \rangle$ be the free unital associative $F$-algebra on the free generating set $X = \{ x_1, x_2, \dots \}$. A subalgebra (a vector subspace) $V$ in $F \langle X \rangle$ is called a $T$-subalgebra (a $T$-subspace) if $\phi (V) \subseteq V$ for all endomorphisms $\phi$ of $F \langle X \rangle$. For an algebra $G$, its central polynomials form a $T$-subalgebra $C(G)$ in $F \langle X \rangle$. Over a field of characteristic $p > 2$ there are algebras $G$ whose algebras of all central polynomials $C (G)$ are not finitely generated as $T$-subspaces in $F \langle X \rangle$. However, no example of an algebra $G$ such that $C(G)$ is not finitely generated as a $T$-subalgebra is known yet. In the present paper we construct the first example of a $2$-graded unital associative algebra $B$ over a field of characteristic $p>2$ whose algebra $C_2 (B)$ of all $2$-graded central polynomials is not finitely generated as a $T_2$-subalgebra in the free $2$-graded unital associative $F$-algebra $F \langle Y,Z \rangle$. Here $Y = \{ y_1, y_2, \dots \}$ and $Z = \{ z_1, z_2, \dots \}$ are sets of even and odd free generators of $F \langle Y,Z \rangle$, respectively. We hope that our example will help to construct an algebra $G$ whose algebra $C(G)$ of (ordinary) central polynomials is not finitely generated as a $T$-subalgebra in $F \langle X \rangle$.Comment: 8 page

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The Giotto magnetometer experiment employs a lowmass (1.357 kg), low-power (818 mW) instrument in a dual magnetometer configuration using flux gate sensors of the ring core type. It has provided accurate vector magnetic field measurements on its way to and near comet Halley, working flawlessly from switch-on on 22 August 1985 to the formal end of the mission on 15 March 1986.