Some results in noncommutative ring theory

Smoktunowicz, Agata

doi:10.4171/022-2/12

Cited by 9 publications

(7 citation statements)

References 28 publications

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“…It was shown by Amitsur in 1973 that if R is a nil algebra over an uncountable field then polynomial rings in many commuting variables over R are also nil [7,9]. However in general polynomial rings over nil rings need not be nil [20,21]. Our next result shows that polynomial rings over some nil rings contain noncommutative free algebras of rank two, and hence are very far from being nil.…”

Section: Introductionmentioning

confidence: 64%

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Makar-Limanov's conjecture on free subalgebras

Smoktunowicz

2009

Advances in Mathematics

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It is proved that over every countable field K there is a nil algebra R such that the algebra obtained from R by extending the field K contains noncommutative free subalgebras of arbitrarily high rank.It is also shown that over every countable field K there is an algebra R without noncommutative free subalgebras of rank two such that the algebra obtained from R by extending the field K contains a noncommutative free subalgebra of rank two. This answers a question of Makar-Limanov [15]. Mathematics Subject Classification MSC2000: 16S10, 16N40, 16W50, 16U99Key words: free subalgebras, extensions of algebras, nil rings nilpotent matrices 2 [1,4,8,11,13,14,17,18,22]. In the influential paper of Makar-Limanov [12] several interesting open questions have been asked. In particular Makar-Limanov conjectured that if R is a finitely generated infinite dimensional algebraic division algebra then R contains a free subalgebra in two generators.Another question along this line was asked by Anick [1] in mid 1980's: Let R be a finitely presented algebra with exponential growth. Does it follow that R contains a free subalgebra in two generators? In the same paper he shown that finitely presented monomial algebras with exponential growth contain free subalgebras in two generators [1]. In [13] Makar-Limanov proved that the quotient algebra of the Weyl algebra contains a free subalgebra in two generators. He also conjectured that the following holds. Conjecture 1.1 (Makar-Limanov, [15], [2])If R is an algebra without free subalgebras of rank two and S is an extensions of R obtained by extending the field K then S doesn't contain a free Kalgebra of rank two.Makar-Limanov mentioned that the truth of this conjecture would imply that we need only to consider algebras over uncountable fields in his mentioned above conjecture on the division algebras [12]. Conjecture 1.1 in the case of skew-fields, as stated in [12], attracted a lot of attention and is known to be true in several important cases [3,4,5,6,10,14,17,19]. In 1996 Reichstein showed that Conjecture 1.1 holds for algebras over uncountable fields [16].The purpose of this paper is to show that the situation is completely different for algebras over countable fields, as shown in the next theorem.Theorem 1.1 Over every countable field K there is an algebra A without free noncommutative subalgebras of rank two such that the polynomial ring A[x] in one indeterminate x over A contains a free noncommutative Kalgebra of rank two.Note that if an algebra contains a noncommutative free algebra of rank two then it also contains a noncommutative free algebra of arbitrarily high rank.As an application the following result is obtained. nilpotent matrices

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Section: Introductionmentioning

confidence: 64%

“…There are connections with problems in nil rings. A nil element is obviously algebraic, and in the converse direction, it is possible to construct an associated graded algebra connected with an algebraic algebra in such a way that the positive part is a graded nil algebra [21].…”

Section: Introductionmentioning

confidence: 99%

Makar-Limanov's conjecture on free subalgebras

Smoktunowicz

2009

Advances in Mathematics

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“…It is not known whether the Jacobson radical of a finitely presented algebra always is a nil ideal; see [15] and [3,Problem 1.92]. This is a problem attributed to Amitsur and Latyshev.…”

Section: Problemmentioning

confidence: 99%

On the semiprimitivity of finitely generated algebras

Okniński

2014

Proc. Amer. Math. Soc.

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“…This question is very important to clarify in the light of key problems in the ring theory concerned with the behaviour of nilpotent elements. Examples of such problems are the problem on the existence of simple nil ring, solved in the affirmative in [8], the Köthe conjecture, the Burnside-type problem for finitely presented rings [9].…”

Section: Introductionmentioning

confidence: 99%

“…Examples of such problems are the problem on the existence of simple nil ring, solved in affirmative by A.Smoktuniwicz [8], the Köthe conjecture, the Burnside type problem for finitely presented rings (see [9]). …”

Section: Introductionmentioning

confidence: 99%

The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

Iyudu

Shkarin

2011

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We study the question on whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper 'Generic algebras and CWcomplexes', Princeton Univ. Press., where he proved that the estimate is attained for the number of quadratic relations d 2 , and conjectured that it is the case for any number of quadratic relations. The particular point where the number of relations is equal to n(n−1) 2 was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional.We prove that over any infinite field, the Anick conjecture holds for d 4(n 2 +n) 9and arbitrary number of generators, and confirm the Vershik conjecture over any field of characteristic 0. We give also a series of related asymptotic results.

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Some results in noncommutative ring theory

Abstract: Abstract. In this paper we survey some results on the structure of noncommutative rings. We focus particularly on nil rings, Jacobson radical rings and rings with finite Gelfand-Kirillov dimension.

Cited by 9 publications

References 28 publications

Makar-Limanov's conjecture on free subalgebras

Makar-Limanov's conjecture on free subalgebras

On the semiprimitivity of finitely generated algebras

The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

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