The additive group of a Lie nilpotent associative ring

Krasilnikov, Alexei

doi:10.1016/j.jalgebra.2013.06.021

Cited by 17 publications

(34 citation statements)

References 12 publications

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“…(see [14,Theorem 1.1]) so for m = 3, n = 2 Theorem 1.1 cannot be improved. On the other hand, if m = n = 3 then, for each associative ring A and all a i , b j ∈ A,…”

Section: Introductionmentioning

confidence: 99%

“…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%

“…, b n ]+T (m+n−1) are non-trivial elements of H ⊂ Z X /T (m+n−1) whose order, by Theorem 1.1, is equal to 3 (if (m, n) = (3, 3)). If m = 3, n = 2 then such products of commutators generate H as a two-sided ideal in Z X /T (4) (see [7,14]). One might expect a similar situation if k > 5.…”

Section: Introductionmentioning

confidence: 99%

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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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“…(see [14,Theorem 1.1]) so for m = 3, n = 2 Theorem 1.1 cannot be improved. On the other hand, if m = n = 3 then, for each associative ring A and all a i , b j ∈ A,…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Some products of commutators in an associative ring

Deryabina

Krasilnikov

2019

Int. J. Algebra Comput.

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“…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]). The aim of the present article is to exhibit a similar generating set for T (5) .…”

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%

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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Identities in Group Rings, Enveloping Algebras and Poisson Algebras

Petrogradsky

2020

Springer INdAM Series

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The additive group of a Lie nilpotent associative ring

Cited by 17 publications

References 12 publications

Some products of commutators in an associative ring

Some products of commutators in an associative ring

Relations in universal Lie nilpotent associative algebras of class 4

Identities in Group Rings, Enveloping Algebras and Poisson Algebras

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