67 results on '"Agata Smoktunowicz"'
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2. On skew braces and their ideals.
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Olexandr Konovalov, Agata Smoktunowicz, and Leandro Vendramin
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- 2021
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3. Erratum to the Paper 'On Skew Braces and Their Ideals'.
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Olexandr Konovalov, Agata Smoktunowicz, and Leandro Vendramin
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- 2022
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4. The three-term recursion for Chebyshev polynomials is mixed forward-backward stable.
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Alicja Smoktunowicz, Agata Smoktunowicz, and Ewa Pawelec
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- 2015
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5. From braces to Hecke algebras and quantum groups
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Anastasia Doikou and Agata Smoktunowicz
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Algebra and Number Theory ,Mathematics::Quantum Algebra ,Applied Mathematics ,Mathematics - Quantum Algebra ,Mathematics - Rings and Algebras ,Mathematics - Group Theory ,Mathematical Physics - Abstract
We examine links between the theory of braces and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras and we identify new quantum groups associated to set-theoretic solutions coming from braces. We also construct a novel class of quantum discrete integrable systems and we derive symmetries for the corresponding periodic transfer matrices., Comment: 25 pages, LaTex. Clarifying comments added, a few typos corrected. E-pub ahead of print in: Journal of Algebra and its Applications
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- 2022
6. On the passage from finite braces to pre-Lie rings
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Agata Smoktunowicz
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Mathematics::Group Theory ,17A65, 17D99, 20F18, 20F40 ,Rings and Algebras (math.RA) ,General Mathematics ,Mathematics::Rings and Algebras ,FOS: Mathematics ,Mathematics - Rings and Algebras ,Mathematics::Representation Theory - Abstract
Let p be a prime number. We show that there is a one-to-one correspondence between the set of strongly nilpotent braces and the set of nilpotent pre-Lie rings of cardinality $p^{n}$, for sufficiently large p. Moreover, there is an injective mapping from the set of left nilpotent pre-Lie rings into the set of left nilpotent braces of cardinality $p^{n}$ for n+1, Comment: arXiv admin note: text overlap with arXiv:2011.07611 Some assumptions were removed from Lemma 15 answering Question 3, also Question 3 was removed.A gap in the proof of Theorem 2 was corrected
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- 2022
7. Erratum to the Paper 'On Skew Braces and Their Ideals'
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Agata Smoktunowicz, Alexander Konovalov, Leandro Vendramin, Faculty of Law and Criminology, and Mathematics
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Algebra ,General Mathematics ,Skew ,Mathematics - Abstract
Erratum to the paper [Konovalov, Alexander; Smoktunowicz, Agata; Vendramin, Leandro. On skew braces and their ideals. Exp. Math. 30 (2021), no. 1, 95–104. DOI: 10.1080/10586458.2018.1492476.].
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- 2022
8. Five solved problems on radicals of Ore extensions
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Michał Ziembowski, Be'eri Greenfeld, and Agata Smoktunowicz
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Power series ,Pure mathematics ,skew-polynomial extensions ,General Mathematics ,Polynomial ring ,differential polynomial rings ,Jacobson radical ,graded nil algebras ,01 natural sciences ,Skew-polynomial extensions ,0101 mathematics ,Mathematics ,16N60 ,16N80 ,Conjecture ,16N20 ,16N40 ,010102 general mathematics ,Graded nil algebras ,Graded ring ,Locally nilpotent ,Skew ,Differential polynomial rings ,16S36 ,In degree ,16S32 ,16W50 - Abstract
The first named author was partially supported by an ISF grant #1623/16. The second named author was supported by ERC Advanced grant Coimbra 320974. The third named author was supported by the Polish National Science Centre grant UMO2017/25/B/ST1/00384. We answer several open questions and establish new results concerningdierential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results. If R is prime radical and δ is a derivation of R, then the dierential polynomial ring R[X; δ] is locally nilpotent. This answers an open question posed in [41]. The nil radical of a dierential polynomial ring R[X; δ] takes the form I[X; δ] for some ideal I of R, provided that the base field is infinite. This answers an open question posed in [30] for algebras over infinite fields. If R is a graded algebra generated in degree 1 over a field of characteristic zero and δ is a grading preserving derivation on R, then the Jacobson radical of R is δ-stable. Examples are given to show the necessity of all conditions, thereby proving this result is sharp. Skew polynomial rings with natural grading are locally nilpotent if and only if they are graded locally nilpotent. The power series ring R[[X; σ; δ]] is well-defined whenever δ is a locally nilpotent σ-derivation; this answers a conjecture from [13], and opens up the possibility of generalizing many research directions studied thus far only when further restrictions are put on δ.
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- 2019
9. Some Braces of Cardinality $p^{4}$ and Related Hopf-Galois Extensions
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dora puljic, Agata Smoktunowicz, and Kayvan Nejabati Zenouz
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Mathematics::Group Theory ,Mathematics::K-Theory and Homology ,Rings and Algebras (math.RA) ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,Mathematics::Rings and Algebras ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Mathematics - Rings and Algebras ,Group Theory (math.GR) ,QA ,Mathematics - Group Theory - Abstract
We describe all Fp-braces of cardinality p4 which are not\ud right nilpotent. The constructed braces are solvable and prime and\ud contain a non-zero strongly nilpotent ideal. We use the constructed\ud braces to construct examples of finitely dimensional pre-Lie algebras\ud which are left nilpotent but not right nilpotent. We also explain some\ud well known results about the correspondence between braces and Hopf-Galois extensions using the notion of Hopf-Galois extensions associated to a given brace. This can be applied to the constructed Fp-braces.
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- 2021
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10. Algebraic approach to Rump’s results on relations between braces and pre-Lie algebras
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Agata Smoktunowicz
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Pure mathematics ,Algebra and Number Theory ,Rump ,Applied Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,01 natural sciences ,Nilpotent ,Lie algebra ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right [Formula: see text]-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors, works locally. In this paper, we explain Rump’s correspondence using only algebraic formulae. An algebraic interpretation of the correspondence works for fields of sufficiently large prime characteristic as well as for fields of characteristic zero.
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- 2020
11. Set-theoretic solutions of the Yang–Baxter equation and new classes of R-matrices
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Alicja Smoktunowicz and Agata Smoktunowicz
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Numerical Analysis ,Pure mathematics ,Algebra and Number Theory ,Yang–Baxter equation ,010102 general mathematics ,010103 numerical & computational mathematics ,Construct (python library) ,01 natural sciences ,Unitary state ,Set (abstract data type) ,Singular value ,Mathematics::Quantum Algebra ,Linear algebra ,Discrete Mathematics and Combinatorics ,Geometry and Topology ,0101 mathematics ,Indecomposable module ,Quantum ,Mathematics - Abstract
We describe several methods of constructing R-matrices that are dependent upon many parameters, for example unitary R-matrices and R-matrices whose entries are functions. As an application, we construct examples of R-matrices with prescribed singular values. We characterise some classes of indecomposable set-theoretic solutions of the quantum Yang–Baxter equation (QYBE) and construct R-matrices related to such solutions. In particular, we establish a correspondence between one-generator braces and indecomposable, non-degenerate involutive set-theoretic solutions of the QYBE, showing that such solutions are abundant. We show that R-matrices related to involutive, non-degenerate solutions of the QYBE have special form. We also investigate some linear algebra questions related to R-matrices.
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- 2018
12. An affine prime non-semiprimitive monomial algebra with quadratic growth.
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Agata Smoktunowicz and Uzi Vishne
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- 2006
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13. Set theoretic Yang-Baxter & reflection equations and quantum group symmetries
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Anastasia Doikou and Agata Smoktunowicz
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Pure mathematics ,Reflection formula ,Hecke algebra ,Quantum group ,010102 general mathematics ,Subalgebra ,Duality (mathematics) ,FOS: Physical sciences ,Boundary (topology) ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Mathematics - Rings and Algebras ,01 natural sciences ,Reflection (mathematics) ,Rings and Algebras (math.RA) ,0103 physical sciences ,Homogeneous space ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,010307 mathematical physics ,0101 mathematics ,Mathematical Physics ,Mathematics - Abstract
Connections between set-theoretic Yang-Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic $R$-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for $R$-matrices being Baxterized solutions of the $A$-type Hecke algebra ${\cal H}_N(q=1)$. We show in the case of the reflection algebra that there exists a ``boundary'' finite sub-algebra for some special choice of ``boundary'' elements of the $B$-type Hecke algebra ${\cal B}_N(q=1, Q)$. We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the $B$-type Hecke algebra. This is one of the fundamental results of this investigation together with the proof of the duality between the boundary finite subalgebra and the $B$-type Hecke algebra. These are universal statements that largely generalize previous relevant findings, and also allow the investigation of the symmetries of the double row transfer matrix., 38 pages, Latex. Various clarifying comments introduced, two references added. Version to appear in LMP
- Published
- 2020
14. A note on set-theoretic solutions of the Yang–Baxter equation
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Agata Smoktunowicz
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Semidirect product ,Ring (mathematics) ,Algebra and Number Theory ,Yang–Baxter equation ,Mathematics::Rings and Algebras ,010102 general mathematics ,Mathematics - Rings and Algebras ,Jacobson radical ,Permutation group ,01 natural sciences ,Brace ,Combinatorics ,Cardinality ,Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,Wreath product ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper shows that every finite non-degenerate involutive set theoretic solution (X,r) of the Yang-Baxter equation whose symmetric group has cardinality which a cube-free number is a multipermutation solution. Some properties of finite braces are also investigated (Theorems 3, 5 and 11). It is also shown that if A is a left brace whose cardinality is an odd number and (-a) b=-(ab) for all a, b A, then A is a two-sided brace and hence a Jacobson radical ring. It is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a finite multipermutation level., Comment: Added a missing assumption in Theorem 5.2
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- 2018
15. How far can we go with Amitsur’s theorem in differential polynomial rings?
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Agata Smoktunowicz
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Principal ideal ring ,Noncommutative ring ,Mathematics::Commutative Algebra ,General Mathematics ,Polynomial ring ,Mathematics::Rings and Algebras ,010102 general mathematics ,010103 numerical & computational mathematics ,Jacobson radical ,01 natural sciences ,Matrix polynomial ,Combinatorics ,Minimal polynomial (field theory) ,Nil ideal ,Von Neumann regular ring ,0101 mathematics ,Mathematics - Abstract
A well-known theorem by S. A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x;D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x;D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p > 0. The main idea of the proof introduces a new way of embedding differential polynomial rings into bigger rings, which we name platinum rings, plus a key part of the proof involves the solution of matrix theory-based problems.
- Published
- 2017
16. On skew braces and their ideals
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Alexander Konovalov, Agata Smoktunowicz, Leandro Vendramin, EPSRC, European Commission, University of St Andrews. School of Computer Science, University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra, and Mathematics
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Pure mathematics ,Braces ,Mathematics::Commutative Algebra ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Mathematics::Rings and Algebras ,Skew ,Semiprime ring ,Yang-Baxter equation ,DAS ,0102 computer and information sciences ,Radical rings ,01 natural sciences ,Prime (order theory) ,010201 computation theory & mathematics ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,AGATA ,QA Mathematics ,0101 mathematics ,QA ,Mathematics - Abstract
The first-named author is partially supported by CCP CoDiMa (EP/M022641/1) and the OpenDreamKit Horizon 2020 European Research Infrastructures project (#676541). The second-named author is supported by the ERC Advanced grant 320974. The third-named author is supported by PICT-201-0147, MATH-AmSud 17MATH-01 and ERC Advanced grant 320974. We define combinatorial representations of finite skew braces and use this idea to produce a database of skew braces of small size. This database is then used to explore different concepts of the theory of skew braces such as ideals, series of ideals, prime and semiprime ideals, Baer and Wedderburn radicals and solvability. The paper contains several questions. Postprint
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- 2018
17. Skew left braces of nilpotent type
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Agata Smoktunowicz, Ferran Cedó, Leandro Vendramin, and Mathematics
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Left and right ,Mathematics(all) ,Pure mathematics ,General Mathematics ,Structure (category theory) ,010103 numerical & computational mathematics ,Group Theory (math.GR) ,Type (model theory) ,01 natural sciences ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,0101 mathematics ,16T25 (primary) ,Mathematics ,Series (mathematics) ,010102 general mathematics ,Mathematics::Rings and Algebras ,Skew ,81R50 (secondary) ,Mathematics - Rings and Algebras ,Nilpotent ,Rings and Algebras (math.RA) ,Indecomposable module ,Mathematics - Group Theory - Abstract
We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang-Baxter equation are explored using the structure of skew left braces., Comment: 27 pages. Accepted for publication in Proc. London Math. Soc. (3)
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- 2018
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18. Combinatorial solutions to the reflection equation
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Robert Weston, Agata Smoktunowicz, Leandro Vendramin, and Mathematics
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Reflection formula ,brace ,FOS: Physical sciences ,Group Theory (math.GR) ,01 natural sciences ,Mathematics - Quantum Algebra ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,FOS: Mathematics ,Applied mathematics ,Quantum Algebra (math.QA) ,0101 mathematics ,Mathematical Physics ,Mathematics ,Algebra and Number Theory ,Reflection equation ,Yang–Baxter equation ,010102 general mathematics ,Skew ,Radical ring ,Mathematics - Rings and Algebras ,Construct (python library) ,Mathematical Physics (math-ph) ,Brace ,yang-baxter equation ,Skew brace ,Rings and Algebras (math.RA) ,010307 mathematical physics ,Mathematics - Group Theory ,Parameter dependent - Abstract
We use ring-theoretic methods and methods from the theory of skew braces to produce set-theoretic solutions to the reflection equation. We also use set-theoretic solutions to construct solutions to the parameter-dependent reflection equation., Comment: 20 pages. Final version
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- 2018
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19. Infinite dimensional, affine nil algebras A⊗Aop and A⊗A exist
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Agata Smoktunowicz
- Subjects
Discrete mathematics ,Algebra and Number Theory ,010102 general mathematics ,010103 numerical & computational mathematics ,Affine transformation ,Finitely-generated abelian group ,0101 mathematics ,Algebraically closed field ,Algebra over a field ,01 natural sciences ,Mathematics - Abstract
In this paper we answer two questions posed by Puczylowski in 1993, by constructing, over any algebraically closed field K, a finitely generated, infinite dimensional algebra A such that the algebras A ⊗ K A and A ⊗ K A op are nil.
- Published
- 2015
20. Chains of Prime Ideals and Primitivity of ℤ $\mathbb {Z}$ -Graded Algebras
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André Leroy, Agata Smoktunowicz, Be'eri Greenfeld, and Michał Ziembowski
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Quadratic growth ,Discrete mathematics ,Pure mathematics ,Tensor product ,General Mathematics ,Dimension (graph theory) ,Gelfand–Kirillov dimension ,Krull dimension ,Affine transformation ,Algebra over a field ,Prime (order theory) ,Mathematics - Abstract
In this paper we provide some results regarding affine, prime, \(\mathbb {Z}\)-graded algebras \(R=\bigoplus _{i\in \mathbb {Z}}R_{i}\) generated by elements with degrees 1,−1 and 0, with R0 finite-dimensional. The results are as follows. These algebras have a classical Krull dimension when they have quadratic growth. If Rk≠0 for almost all k then R is semiprimitive. If in addition R has GK dimension less than 3 then R is either primitive or PI. The tensor product of an arbitrary Brown-McCoy radical algebra of Gelfand Kirillov dimension less than three and any other algebra is Brown-McCoy radical.
- Published
- 2015
21. Braces and symmetric groups with special conditions
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Ferran Cedó, Tatiana Gateva-Ivanova, and Agata Smoktunowicz
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Pure mathematics ,Ring (mathematics) ,Algebra and Number Theory ,Group (mathematics) ,010102 general mathematics ,Structure (category theory) ,Jacobson radical ,01 natural sciences ,Brace ,Symmetric group ,Bounded function ,0103 physical sciences ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Order (group theory) ,Quantum Algebra (math.QA) ,16T25, 16W22, 16N20, 16N40, 20F16, 81R50 ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
We study symmetric groups and left braces satisfying special conditions, or identities. We are particularly interested in the impact of conditions like $\textbf{Raut}$ and $\textbf{lri}$ on the properties of the symmetric group and its associated brace. We show that the symmetric group $G=G(X,r)$ associated to a nontrivial solution $(X,r)$ has multipermutation level $2$ if and only if $G$ satisfies $\textbf{lri}$. In the special case of a two-sided brace we express each of the conditions $\textbf{lri}$ and $\textbf{Raut}$ as identities on the associated radical ring $G_*$. We apply these to construct examples of two-sided braces satisfying some prescribed conditions. In particular we construct a finite two-sided brace with condition $\textbf{Raut}$ which does not satisfy $\textbf{lri}$. (It is known that condition $\textbf{lri}$ implies $\textbf{Raut}$). We show that a finitely generated two-sided brace which satisfies \textbf{lri} has a finite multipermutation level which is bounded by the number of its generators., Comment: 20 pages
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- 2017
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22. On the Yang-Baxter equation and left nilpotent left braces
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Tatiana Gateva-Ivanova, Ferran Cedó, and Agata Smoktunowicz
- Subjects
Pure mathematics ,Algebra and Number Theory ,Group (mathematics) ,Yang–Baxter equation ,010102 general mathematics ,Structure (category theory) ,Group Theory (math.GR) ,01 natural sciences ,Brace ,Nilpotent ,Chain (algebraic topology) ,Mathematics::Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Mathematics - Group Theory ,Mathematics ,Z-group ,Engel group - Abstract
We study non-degenerate involutive set-theoretic solutions ( X , r ) of the Yang–Baxter equation , we call them solutions. We prove that the structure group G ( X , r ) of a finite non-trivial solution ( X , r ) cannot be an Engel group. It is known that the structure group G ( X , r ) of a finite multipermutation solution ( X , r ) is a poly- Z group, thus our result gives a rich source of examples of braided groups and left braces G ( X , r ) which are poly- Z groups but not Engel groups. We find an explicit relation between the multipermutation level of a left brace and the length of the radical chain A ( n + 1 ) = A ( n ) ⁎ A introduced by Rump. We also show that a finite solution of the Yang–Baxter equation can be embedded in a convenient way into a finite left brace, or equivalently into a finite involutive braided group.
- Published
- 2017
23. Golod-Shafarevich type theorems and potential algebras
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Natalia Iyudu and Agata Smoktunowicz
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Pure mathematics ,Polynomial ,General Mathematics ,FOS: Physical sciences ,Koszul complex ,Group Theory (math.GR) ,01 natural sciences ,Gröbner basis ,symbols.namesake ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,0101 mathematics ,Representation Theory (math.RT) ,Mathematical Physics ,Hilbert–Poincaré series ,Mathematics ,Conjecture ,Degree (graph theory) ,010102 general mathematics ,Mathematics - Rings and Algebras ,Mathematical Physics (math-ph) ,Noncommutative geometry ,Minimal model program ,Rings and Algebras (math.RA) ,symbols ,Mathematics - Group Theory ,Mathematics - Representation Theory - Abstract
Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gr\"obner basis theory and generalized Golod-Shafarevich type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gr\"obner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than $8$. This answers a question of Wemyss \cite{Wemyss}, related to the geometric argument of Toda \cite{T}. We derive from the improved version of the Golod-Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove, that potential algebra for any homogeneous potential of degree $n\geq 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class ${\cal P}_n$ of potential algebras with homogeneous potential of degree $n+1\geq 4$, the minimal Hilbert series is $H_n=\frac{1}{1-2t+2t^n-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but non-linear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar-Vafa invariants., Comment: To appear in IMRN
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- 2017
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24. Growth, entropy and commutativity of algebras satisfying prescribed relations
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Agata Smoktunowicz
- Subjects
Golod-Shaferevich algebras ,General Mathematics ,Non-associative algebra ,POWER-SERIES RINGS ,General Physics and Astronomy ,BEZOUT ,01 natural sciences ,Quadratic algebra ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Commutative property ,Mathematics ,Discrete mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,16P40, 16S15, 16W50, 16P90 ,GELFAND-KIRILLOV DIMENSION ,Mathematics - Rings and Algebras ,Noncommutative geometry ,Growth of algebras and the Gelfand-Kirillov dimension ,Rings and Algebras (math.RA) ,Gelfand–Kirillov dimension ,Uncountable set ,Gravitational singularity ,010307 mathematical physics ,Nest algebra - Abstract
In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfy some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod-Shafarevich algebras. This paper provides bounds for the growth function on images of Golod-Shafarevich algebras based upon the number of defining relations. This extends results from [32], [33]. Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky [7] by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov [40]. Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss in [8]., Comment: arXiv admin note: text overlap with arXiv:1207.6503
- Published
- 2014
25. JACOBSON RADICAL ALGEBRAS WITH QUADRATIC GROWTH
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Alexander Young and Agata Smoktunowicz
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Quadratic growth ,Discrete mathematics ,Mathematics Subject Classification ,General Mathematics ,Existential quantification ,Gelfand–Kirillov dimension ,Countable set ,GELFAND-KIRILLOV DIMENSION ,Jacobson radical ,Finitely-generated abelian group ,Algebraically closed field ,NIL ALGEBRAS ,Mathematics - Abstract
We show that over every countable algebraically closed field $\mathbb{K}$ there exists a finitely generated $\mathbb{K}$-algebra that is Jacobson radical, infinite-dimensional, generated by two elements, graded and has quadratic growth. We also propose a way of constructing examples of algebras with quadratic growth that satisfy special types of relations.
- Published
- 2013
26. On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation
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Agata Smoktunowicz
- Subjects
Multiplicative group ,Group (mathematics) ,Direct sum ,Yang–Baxter equation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Rings and Algebras ,Mathematics - Rings and Algebras ,Engel group, nilpotent group, adjoint group of a ring, braces, nil rings, nil algebras, the Yang-Baxter equation ,01 natural sciences ,Section (fiber bundle) ,Combinatorics ,Nilpotent ,Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Nilpotent group ,Engel group ,Mathematics - Abstract
It is shown that over an arbitrary field there exists a nil algebra $R$ whose adjoint group $R^{o}$ is not an Engel group. This answers a question by Amberg and Sysak from 1997 [5] and answers related questions from [3, 44]. The case of an uncountable field also answers a recent question by Zelmanov. In [38], Rump introduced braces and radical chains $A^{n+1}=A\cdot A^{n}$ and $A^{(n+1)}=A^{(n)}\cdot A$ of a brace $A$. We show that the adjoint group $A^{o}$ of a finite right brace is a nilpotent group if and only if $A^{(n)}=0$ for some $n$. We also show that the adjoint group of $A^{o}$ of a finite left brace $A$ is a nilpotent group if and only if $A^{n}=0$ for some $n$. Moreover, if $A^{o}$ is a nilpotent group then $A$ is the direct sum of braces whose cardinatities are powers of prime numbers. Notice that $A^{o}$ is sometimes called the multiplicative group of a brace $A$ (for example in [13]). We also introduce a chain of ideals $A^{[n]}$ of a left brace $A$ and then use it to investigate braces which satisfy $A^{n}=0$ and $A^{(m)}=0$ for some $m, n$ (Theorems 2, 3). In Section 2 we describe connections between our results and braided groups and the Yang-Baxter equation. It is worth noticing that by a result by Gateva-Ivanova [17] braces are in one-to-one correspondence with braided groups with involutive braided operators., To appear in the Transactions of the AMS. Improved the presentation, corrected a few typos
- Published
- 2015
27. Jacobson radical non-nil algebras of Gel’fand-Kirillov dimension 2
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Laurent Bartholdi and Agata Smoktunowicz
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Mathematics(all) ,Pure mathematics ,Conjecture ,General Mathematics ,Mathematics::Rings and Algebras ,Dimension (graph theory) ,Field (mathematics) ,Jacobson radical ,Prime (order theory) ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Gelfand–Kirillov dimension ,Countable set ,Computer Science::Symbolic Computation ,Physics::Chemical Physics ,16N40, 16P90 ,math.RA ,Associative property ,Mathematics - Abstract
For an arbitrary countable field, we construct an associative algebra that is graded, generated by two elements is Jacobson radical, is not nil, is prime, is not PI, and has Gelfand-Kirillov dimension two. This refutes a conjecture attributed to Goodearl. The Jacobson radical is very important for the study of noncommutative algebras. For a given ring R one usually studies the Jacobson radical J(R) of R, and the semiprimitive part R/J(R). As related evidence of a connection between these notions, a result of Amitsur says that the Jacobson radical of a finitely generated algebra over an uncountable field is nil, and it is known that all nil rings are Jacobson radical. It is important to know when Jacobson radical are nil because nil rings have interesting properties. For example subalgebras of nil algebras are nil, which does not hold in general for Jacobson radical rings. The Jacobson radical is important for determining the structure of rings and is a generalization of the Wedderburn radical for finitely dimensional algebras.
- Published
- 2012
28. Nil algebras with restricted growth
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Alexander Young, T. H. Lenagan, and Agata Smoktunowicz
- Subjects
Finitely generated algebra ,Pure mathematics ,General Mathematics ,Existential quantification ,Mathematics::Rings and Algebras ,Field (mathematics) ,Mathematics - Rings and Algebras ,Dimension (vector space) ,Mathematics Subject Classification ,Rings and Algebras (math.RA) ,Gelfand–Kirillov dimension ,FOS: Mathematics ,Countable set ,16N40, 16P90 ,Mathematics::Representation Theory ,Mathematics - Abstract
It is shown that over an arbitrary countable field, there exists a finitely generated algebra that is nil, infinite dimensional, and has Gelfand-Kirillov dimension at most three., 20 pages
- Published
- 2012
29. Primitive algebraic algebras of polynomially bounded growth
- Author
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Lance W. Small, Agata Smoktunowicz, Jason P. Bell, Ara, P, Brown, K. A., Lenagan, T. H., Letzter, E. S., Stafford, J. T., and Zhang, J. J.
- Subjects
Pure mathematics ,Existential quantification ,010102 general mathematics ,Dimension (graph theory) ,Field (mathematics) ,Mathematics - Rings and Algebras ,01 natural sciences ,Rings and Algebras (math.RA) ,Bounded function ,0103 physical sciences ,FOS: Mathematics ,Countable set ,010307 mathematical physics ,Finitely-generated abelian group ,0101 mathematics ,Algebraic number ,math.RA ,Mathematics - Abstract
We show that if $k$ is a countable field, then there exists a finitely generated, infinite-dimensional, primitive algebraic $k$-algebra $A$ whose Gelfand-Kirillov dimension is at most six. In addition to this we construct a two-generated primitive algebraic $k$-algebra. We also pose many open problems., 12 pages
- Published
- 2012
30. Makar-Limanov's conjecture on free subalgebras
- Author
-
Agata Smoktunowicz
- Subjects
Mathematics(all) ,Rank (linear algebra) ,General Mathematics ,Polynomial ring ,Free subalgebras ,Field (mathematics) ,FREE SUBGROUPS ,DIVISION RINGS ,Combinatorics ,Mathematics::Algebraic Geometry ,Mathematics::K-Theory and Homology ,Free algebra ,Mathematics::Quantum Algebra ,FOS: Mathematics ,16S10, 16N40, 16W50 ,Countable set ,ALGEBRAS ,Mathematics ,Discrete mathematics ,FREE SUBSEMIGROUPS ,Mathematics::Commutative Algebra ,Mathematics::Operator Algebras ,Subalgebra ,Mathematics::Rings and Algebras ,Mathematics - Rings and Algebras ,Noncommutative geometry ,Extensions of algebras ,Rings and Algebras (math.RA) ,Nil rings ,POLYNOMIAL-RINGS ,Central simple algebra ,FRACTIONS - Abstract
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 [Lenny Makar-Limanov, private communication, Beijing, June 2007].
- Published
- 2009
- Full Text
- View/download PDF
31. GROWTH OF MODULES OVER GENERIC GOLOD-SHAFAREVICH-ALGEBRAS
- Author
-
Agata Smoktunowicz
- Subjects
Combinatorics ,Stallings theorem about ends of groups ,Exponential growth ,Homogeneous ,General Mathematics ,Field (mathematics) ,Transcendence degree ,Finitely-generated abelian group ,Mathematics - Abstract
Let K be a field of infinite transcendence degree and let A be a finitely generated K-algebra. Suppose that the number of homogeneous generic relations in A of degrees smaller than n grows exponentially with n. Then all infinitely-dimensional finitely generated A-modules have exponential growth. In particular there are Golod-Shafarevich algebras all of whose finitely generated modules either have exponential growth or are finite-dimensional.
- Published
- 2009
32. GK–DIMENSION OF ALGEBRAS WITH MANY GENERIC RELATIONS
- Author
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Agata Smoktunowicz
- Subjects
Algebra ,Mathematics(all) ,General Mathematics ,Mathematics::Rings and Algebras ,Non-associative algebra ,Dimension (graph theory) ,Nest algebra ,Mathematics - Abstract
We prove some results on algebras, satisfying many generic relations. As an application we show that there are Golod–Shafarevich algebras which cannot be homomorphically mapped onto infinite dimensional algebras with finite Gelfand–Kirillov dimension. This answers a question of Zelmanov (Some open problems in the theory of infinite dimensional algebras, J. Korean Math. Soc. 44(5) 2007, 1185–1195).
- Published
- 2009
33. Chains of prime ideals and primitivity of Z-graded algebras
- Author
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Andre Leroy, Be'eri Greenfeld, Agata Smoktunowicz, and Michał Ziembowski
- Published
- 2015
34. The prime spectrum of algebras of quadratic growth
- Author
-
Jason P. Bell and Agata Smoktunowicz
- Subjects
Pure mathematics ,Almost prime ,16P90 ,Mathematics::Number Theory ,EXAMPLES ,Prime decomposition ,Quadratic growth ,primitive rings ,01 natural sciences ,Graded algebra ,Prime (order theory) ,Boolean prime ideal theorem ,Prime factor ,FOS: Mathematics ,0101 mathematics ,Prime power ,Mathematics ,Discrete mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,010102 general mathematics ,GELFAND-KIRILLOV DIMENSION ,Prime element ,Mathematics - Rings and Algebras ,AFFINE ALGEBRAS ,PI rings ,010101 applied mathematics ,Associated prime ,GK dimension ,Rings and Algebras (math.RA) ,quadratic growth ,graded algebra ,Primitive rings - Abstract
We study prime algebras of quadratic growth. Our first result is that if $A$ is a prime monomial algebra of quadratic growth then $A$ has finitely many prime ideals $P$ such that $A/P$ has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the nonzero prime ideals $P$ such that $A/P$ has GK dimension 2 is non-empty, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either primitive or has nonzero locally nilpotent Jacobson radical. Finally, we show that there exists a prime monomial algebra $A$ of GK dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that $A/P$ has GK dimension 1., 23 pages
- Published
- 2008
35. Differential polynomial rings over locally nilpotent rings need not be Jacobson radical
- Author
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Agata Smoktunowicz and Michał Ziembowski
- Subjects
Discrete mathematics ,Pure mathematics ,Algebra and Number Theory ,Noncommutative ring ,Mathematics::Commutative Algebra ,Mathematics::Rings and Algebras ,010102 general mathematics ,Locally nilpotent ,0102 computer and information sciences ,Jacobson radical ,Unipotent ,16. Peace & justice ,01 natural sciences ,Radical of a ring ,Mathematics::Group Theory ,Nilpotent ,010201 computation theory & mathematics ,Nil ideal ,0101 mathematics ,Nilpotent group ,Mathematics - Abstract
We answer a question by Shestakov on the Jacobson radical in differential polynomial rings. We show that if R is a locally nilpotent ring with a derivation D then R [ X ; D ] need not be Jacobson radical. We also show that J ( R [ X ; D ] ) ∩ R is a nil ideal of R in the case where D is a locally nilpotent derivation and R is an algebra over an uncountable field.
- Published
- 2014
- Full Text
- View/download PDF
36. An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension
- Author
-
T. H. Lenagan and Agata Smoktunowicz
- Subjects
Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Locally nilpotent ,Algebra ,Mathematics::Group Theory ,Nilpotent ,Exponential growth ,Mathematics::K-Theory and Homology ,Bounded function ,Gelfand–Kirillov dimension ,Affine transformation ,Algebra over a field ,Mathematics - Abstract
The famous 1960’s construction of Golod and Shafarevich yields infinite dimensional nil, but not nilpotent, algebras. However, these algebras have exponential growth. Here, we construct an infinite dimensional nil, but not locally nilpotent, algebra which has polynomially bounded growth.
- Published
- 2007
37. There are no graded domains with GK dimension strictly between 2 and 3
- Author
-
Agata Smoktunowicz
- Subjects
Discrete mathematics ,Dimension (vector space) ,General Mathematics ,Mathematics::Rings and Algebras ,A domain ,Field (mathematics) ,Finitely-generated abelian group ,Mathematics::Representation Theory ,Mathematics - Abstract
Let K be a field, and let R=⊕n∈NRn be a finitely generated, graded K-algebra which is a domain. It is shown that R cannot have Gelfand-Kirillov dimension strictly between 2 and 3.
- Published
- 2006
38. Centers in domains with quadratic growth
- Author
-
Agata Smoktunowicz
- Subjects
Quadratic growth ,Discrete mathematics ,Polynomial ,16p40 ,General Mathematics ,16s80 ,growth of algebras ,Field (mathematics) ,Center (group theory) ,centers ,Centralizer and normalizer ,Combinatorics ,Gelfand–Kirillov dimension ,16d90 ,QA1-939 ,Division algebra ,the gelfand-kirillov dimension ,Quotient ring ,domains ,16w50 ,Mathematics - Abstract
Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.
- Published
- 2005
39. The Artin-Stafford gap theorem
- Author
-
Agata Smoktunowicz
- Subjects
Applied Mathematics ,General Mathematics ,MathematicsofComputing_GENERAL ,Graded ring ,Graded Lie algebra ,Filtered algebra ,Algebra ,Dimension (vector space) ,Differential graded algebra ,Gelfand–Kirillov dimension ,Gap theorem ,Algebraically closed field ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Mathematics - Abstract
Let K K be an algebraically closed field, and let R R be a finitely graded K K -algebra which is a domain. We show that R R cannot have Gelfand-Kirillov dimension strictly between 2 2 and 3 3 .
- Published
- 2005
40. ARMENDARIZ RINGS AND SEMICOMMUTATIVE RINGS
- Author
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Chan Huh, Agata Smoktunowicz, and Yang Lee
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Mathematics::Rings and Algebras ,Quotient ,Mathematics - Abstract
In this note we concern the structures of Armendariz rings and semicommutative rings which are generalizations of reduced rings, the classical right quotient rings of Armendariz rings, the polynomi...
- Published
- 2002
41. A SIMPLE NIL RING EXISTS
- Author
-
Agata Smoktunowicz
- Subjects
Pure mathematics ,Algebra and Number Theory ,Simple (abstract algebra) ,Ring (chemistry) ,Mathematics - Published
- 2002
42. Amitsur's Conjecture on Polynomial Rings in N Commuting Indeterminates
- Author
-
Agata Smoktunowicz
- Published
- 2002
43. A polynomial ring that is Jacobson radical and not nil
- Author
-
Agata Smoktunowicz and Edmund Puczyłowski
- Subjects
Principal ideal ring ,Reduced ring ,Discrete mathematics ,Noncommutative ring ,Mathematics::Commutative Algebra ,General Mathematics ,Polynomial ring ,Mathematics::Rings and Algebras ,Jacobson radical ,Radical of a ring ,Minimal polynomial (field theory) ,Primitive ring ,Computer Science::Symbolic Computation ,Mathematics - Abstract
In [1] Amitsur conjectured that if a polynomial ring in one indeterminate is Jacobson radical then it is a nil ring. We shall construct an example disproving this conjecture.
- Published
- 2001
44. Polynomial Rings over Nil Rings Need Not Be Nil
- Author
-
Agata Smoktunowicz
- Subjects
Discrete mathematics ,Mathematics::Group Theory ,Algebra and Number Theory ,Mathematics::K-Theory and Homology ,Polynomial ring ,Mathematics::Rings and Algebras ,Mathematics::General Topology ,Countable set ,Field (mathematics) ,Algebra over a field ,Mathematics - Abstract
We construct a nil algebra over a countable field, the polynomial ring over which is not nil. This answers a question of Amitsur.
- Published
- 2000
45. Iterative refinement techniques for solving block linear systems of equations
- Author
-
Alicja Smoktunowicz and Agata Smoktunowicz
- Subjects
Discrete mathematics ,Numerical Analysis ,Floating point ,Applied Mathematics ,Linear system ,Relaxation (iterative method) ,System of linear equations ,law.invention ,Computational Mathematics ,Invertible matrix ,Iterative refinement ,law ,Applied mathematics ,Condition number ,Numerical stability ,Mathematics - Abstract
We study the numerical properties of classical iterative refinement (IR) and k-fold iterative refinement (RIR) for computing the solution of a nonsingular linear system of equations Ax = b with A partitioned into blocks using floating point arithmetic. We assume that all computations are performed in the working (fixed) precision. We prove that the numerical quality of RIR is superior to that of IR.
- Published
- 2013
46. Golod-Shafarevich algebras, free subalgebras and Noetherian images
- Author
-
Agata Smoktunowicz
- Subjects
Noetherian ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Mathematics::Number Theory ,Mathematics::Rings and Algebras ,Mathematics - Rings and Algebras ,Jacobson radical ,Noncommutative geometry ,Prime (order theory) ,Mathematics::Group Theory ,Rings and Algebras (math.RA) ,FOS: Mathematics ,Linear growth ,Mathematics - Abstract
It is shown that Golod–Shafarevich algebras of a reduced number of defining relations contain noncommutative free subalgebras in two generators, and that these algebras can be homomorphically mapped onto prime, Noetherian algebras with linear growth. It is also shown that Golod–Shafarevich algebras of a reduced number of relations cannot be nil.
- Published
- 2013
47. A note on Nil and Jacobson radicals in graded rings
- Author
-
Agata Smoktunowicz
- Subjects
nil rings ,Pure mathematics ,Ring (mathematics) ,Algebra and Number Theory ,Applied Mathematics ,Radical ,010102 general mathematics ,010103 numerical & computational mathematics ,Jacobson radical ,Mathematics - Rings and Algebras ,Subring ,01 natural sciences ,Graded rings ,Radical of a ring ,Rings and Algebras (math.RA) ,Homogeneous ,FOS: Mathematics ,0101 mathematics ,SEMIGROUP RINGS ,nil radical ,Mathematics - Abstract
It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil radicals, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper, it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained.
- Published
- 2013
48. The nil radical of power series rings
- Author
-
Edmund Puczyłowski and Agata Smoktunowicz
- Subjects
Discrete mathematics ,Power series ,Ring (mathematics) ,Pure mathematics ,Mathematics::Commutative Algebra ,Series (mathematics) ,General Mathematics ,Mathematics::Rings and Algebras ,Radical of a ring ,Mathematics::Group Theory ,Nilpotent ,Mathematics::K-Theory and Homology ,Bounded function ,Radical of an ideal ,Ideal (ring theory) ,Mathematics - Abstract
We describe the nil radical of power series rings in non-commuting indeterminates by showing that a series belongs to the radical if and only if the ideal generated by its coefficients is nilpotent. We also show thatt the principal ideals generated by elements of the nil radical of the power series ring in one indeterminate are nil of bounded index.
- Published
- 1999
49. On maximal ideals and the brown-mccoy radical of polynomial rings
- Author
-
Edmund Puczyłowski and Agata Smoktunowicz
- Subjects
Reduced ring ,Principal ideal ring ,Discrete mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Polynomial ring ,Simple ring ,Boolean ring ,Maximal ideal ,Ideal (ring theory) ,Quotient ring ,Mathematics - Abstract
In this paper we obtain some results on maximal ideals of polynomial rings R[x] in one indeterminate x. In particular we complete Ferrero's characterization [5] of rings R with an identity such that R[x] contains a maximal R-disjoint ideal, i.e., a maximal ideal M satisfying M n R = 0. We also get several results on the Brown-McCoy radical of R[x]. Recall that for a given ring A the Brown-McCoy radical U(A) of A is defined as the intersection of all ideals I of A such that A/I is a simple ring with an identity. In particular a ring is Brown-McCoy radical if and only if it cannot be homomorphically mapped onto a ring with an identity, or equivalently, onto a simple ring with an identity. In [7] Krernpa proved that for every ring R, U(R[xJ) = (U(R[x]) n R)[xJ. We shall show that U(R[x]) n R is equal to the intersection of all prime ideals I of R such that the centre of R/I has a non-zero intersection with each non-zero ideal of RII. In particular, if R is a nil ring, then R[x] is Brown-McCoy radical, i.e., R[x] cannot be homomorphically mapped onto
- Published
- 1998
50. Images of Golod-Shafarevich algebras with small growth
- Author
-
Laurent Bartholdi and Agata Smoktunowicz
- Subjects
Quadratic growth ,Discrete mathematics ,Pure mathematics ,Polynomial ,Jordan algebra ,General Mathematics ,Non-associative algebra ,Mathematics - Rings and Algebras ,Function (mathematics) ,Quadratic algebra ,Interior algebra ,Rings and Algebras (math.RA) ,FOS: Mathematics ,Nest algebra ,DIMENSION ,Mathematics - Abstract
We show that Golod-Shafarevich algebras can be homomorphically mapped onto infinite dimensional algebras with polynomial growth when mild assumptions about the number of relations of given degrees are introduced. This answers a question by Zel'manov. In the case where these algebras are finitely presented, we show that they can be mapped onto infinite-dimensional algebras with at most quadratic growth. We then use an elementary construction to show that any sufficiently regular function a parts per thousand(3) n(log n) may occur as the growth function of an algebra.
- Published
- 2011
- Full Text
- View/download PDF
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