On properties of the lower central series of associative algebras

2017

Journal of Algebra

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 m, n ∈ Z, m, n > 1 and at least one of the numbers m, n is odd. Then (1) T (m) T (n) ⊂ T (m+n −1) .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Products of commutators in a Lie nilpotent associative algebra

2017

Journal of Algebra

“…Note that, for a unital associative ring R, we have [2,12,19]). Let R be an arbitrary unital associative and commutative ring such that 1 6 ∈ R and let A be an associative R-algebra. Let m, n ∈ Z, m, n > 1 and at least one of the numbers m, n is odd.…”

Section: Introductionmentioning

confidence: 99%

“…Proposition 1.4 (see [6]). Let R be an arbitrary unital associative and commutative ring such that 1 6 ∈ R and let A be an associative R-algebra. Let k, ℓ be integers such that 0 ≤ ℓ < k. Let m i ≥ 2 (i = 1, .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Products of several commutators in a Lie nilpotent associative algebra

Int. J. Algebra Comput.

2017

Abstract. Let F be a field of characteristic = 2, 3 and let A be a unital associative F -algebra. Define a left-normed commutator [a 1 , a 2 , . .. . , m k be positive integers such that ℓ of them are odd and k − ℓ of them are even. LetThe aim of the present note is to show that, for any positive integers m 1 , . . . , m k , in general,Recently Dangovski has proved that if m 1 , . . . , m k are any positive integers then, in general,

Some products of commutators in an associative ring

Int. J. Algebra Comput.

2019

Let A be a unital associative ring and let T (k) be the two-sided ideal of A generated by all commutators [a1, a2, . . . , a k ] (ai ∈ A) where [a1, a2] = a1a2−a2a1, [a1, . . . , a k−1 , a k ] = [a1, . . . , a k−1 ], a k (k > 2). It has been known that, if either m or n is odd thenfor all ai, bj ∈ A. This was proved by Sharma and Srivastava in 1990 and independently rediscovered later (with different proofs) by various authors. The aim of our note is to give a simple proof of the following result: if at least one of the integers m, n is odd then, for all ai, bj ∈ A, 3 [a1, a2, . . . , am][b1, b2, . . . , bn] ∈ T (m+n−1) .Since it has been known that, in general, [a1, a2, a3][b1, b2] / ∈ T (4) , our result cannot be improved further for all m, n such that at least one of them is odd. AMS MSC Classification: 16R10, 16R40