The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3

Let A be an associative unital algebra, B k its successive quotients of lower central series and N k the successive quotients of ideals generated by lower central series. The geometric and algebraic aspects of B k 's and N k 's have been of great interest since the pioneering work of [11]. In this paper, we will concentrate on the case where A is a noncommutative polynomial algebra over Z modulo a single homogeneous relation. Both the torsion part and the free part of B k 's and N k 's are explored. Many examples are demonstrated in detail, and several general theorems are proved. Finally we end up with Appendix A about the torsion subgroups of N k (A n (Z)) and some open problems.

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“…In fact, Krasilnikov and his coauthors obtained many more results (see [8,6]). For instance, in [8] they proved that…”

Section: Appendix a Torsion Subgroups In N K (A N (Z Z Z))mentioning

confidence: 99%

On lower central series quotients of finitely generated algebras over Z

2015

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“…It was shown in [3] that the additive group of Z X /T (3) is also free abelian. On the other hand, the additive group of the ring Z X /T (4) is a direct sum G ⊕ H of a free abelian group G and an elementary abelian 3-group H (see [7,14]). Computational data by Cordwell, Fei and Zhou presented in [4, Appendix A] suggest that for k ≥ 6 the additive group of the ring Z X /T (k) is also a direct sum of a free abelian group G and a non-trivial elementary abelian 3-group H while for k = 5 this group is free abelian.…”

Section: Introductionmentioning

confidence: 99%

Some products of commutators in an associative ring

Deryabina

Krasilnikov

2019

Int. J. Algebra Comput.

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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

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Section: Introductionmentioning

confidence: 99%

“…It is well-known that T (3) is generated by the polynomials (see, for instance, [6,13,17,23]). If 1 3 ∈ K then a similar generating set for T (4) contains the polynomials of 3 types (see [11,14,24,26]): (1) [ (see [8,Theorem 1.3]). In the latter case, in general, the polynomials [x 1 , x 2 ][x 3 , x 4 , x 5 ] (x i ∈ X) of type (2) do not belong to T (4) but 3 [x 1 , x 2 ][x 3 , x 4 , x 5 ] ∈ T (4) (see [22]).…”

Section: Introductionmentioning

confidence: 99%

“…It was shown in [6] that the additive group of Z X /T (3) is also free abelian. On the other hand, the additive group of the ring Z X /T (4) is a direct sum A ⊕ B of a free abelian group A and an elementary abelian 3-group B (see [22]); bases of A and B were found in [22] and [8], respectively. Computational data by Cordwell, Fei and Zhou presented in [7,Appendix A] suggests that if n ≥ 6 then the additive group of the ring Z X /T (n) is also a direct sum of a free abelian group and a non-trivial elementary abelian 3-group while for n = 5 this group is free abelian.…”

Section: Introductionmentioning

confidence: 99%

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Relations in universal Lie nilpotent associative algebras of class 4

Costa

Krasilnikov

2017

Communications in Algebra

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Abstract. Let K be a unital associative and commutative ring and let K X be the free unital associative K-algebra on a non-empty set X of free generators. Define a left-normed commutator [a1, a2, . . . , an] inductively by [a1, a2] = a1a2 − a2a1, [a1, . . . , an−1, an] = [[a1, . . . , an−1], an] (n ≥ 3). For n ≥ 2, let T (n) be the two-sided ideal in K X generated by all commutators [a1, a2, . . . , an] (ai ∈ K X ).It can be easily seen that the ideal T (2) is generated (as a two-sided ideal in K X ) by the commutators [x1, x2] (xi ∈ X). It is well-known that T (3) is generated by the polynomials [x1, x2, x3] and [x1, x2][x3, x4] + [x1, x3][x2, x4] (xi ∈ X). A similar generating set for T (4) contains 3 types of polynomials in xi ∈ X if 1 3 ∈ K and 5 types if 1 3 / ∈ K. In the present article we exhibit a generating set for T (5) that contains 8 types of polynomials in xi ∈ X. IntroductionLet K be a unital associative and commutative ring and let A be a unital associative K-algebra. Define a left-normed commutator [a 1 , a 2 , . . . , a n ] inductively by [a 1 , a 2 ] = a 1 a 2 −a 2 a 1 , [a 1 , . . . , a n−1 , a n ] = [[a 1 , . . . , a n−1 ], a n ] (n ≥ 3). For n ≥ 2, let T (n) (A) be the two-sided ideal in A generated by all commutators [a 1 , a 2 , . . . , a n ] (a i ∈ A). An algebra A is called Lie nilpotent (of class at most n − 1) if T (n) (A) = 0 for some n ≥ 2.Let K X be the free unital associative K-algebra on a non-empty set X of free generators. Define T (n) = T (n) (K X ). The quotient algebra K X /T (n) can be viewed as the universal Lie nilpotent associative K-algebra of class n − 1 generated by X.The study of Lie nilpotent associative rings and algebras was started by Jennings [18] in 1947. Since then Lie nilpotent associative rings and algebras have been investigated in various papers from various points of view; see, for instance, [2,14,15,16,21,24,25,26] and the bibliography there.Recent interest in Lie nilpotent associative algebras was motivated by the study of the quotients L i /L i+1 of the lower central series of the associated Lie algebra of an associative algebra A; here L n is the linear span in A of the set of all commutators [a 1 , a 2 , . . . , a n ] (a i ∈ A). The study of these quotients L i /L i+1 was initiated in 2007 in a pioneering article of Feigin and Shoikhet [12] for A = C X ; further results on this subject can be found, for example, in [1,3,4,5,6,7,9,10,11,19,20]. Since T (n) (A) is the ideal in A generated by L n , some results about the quotients T (i) (A)/T (i+1) (A) were obtained in these articles as well; in [7,11,19,20] the latter quotients were the primary objects of study.It can be easily seen that the ideal T (2) is generated (as a two-sided ideal in K X ) by the commutators [x 1 , x 2 ] (x i ∈ X). It is well-known that T (3) is generated by the polynomials (see, for instance, [6,13,17,23]). If 1 3 ∈ K then a similar generating set for T (4) contains the polynomials of 3 types (see [11,14,24,26]): (see [8, Theorem 1.3]). In the latter case, in general, the poly...

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The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3

Cited by 8 publications

References 34 publications

On lower central series quotients of finitely generated algebras over Z

On lower central series quotients of finitely generated algebras over Z

Some products of commutators in an associative ring

Relations in universal Lie nilpotent associative algebras of class 4

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