Papers by Agata Smoktunowicz
arXiv (Cornell University), Jul 18, 2020
In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right R-b... more In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right R-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.
Algebras and Representation Theory, Mar 1, 2005
Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indetermin... more Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I[x] for some ideals I of R. All considered rings are associative but not necessarily have identities. Köthe's conjecture states that a ring without nil ideals has no one-sided nil ideals. It is equivalent [4] to the assertion that polynomial rings over nil rings are Jacobson radical. Our main result states that if R is a nil ring and I an ideal in R[x] (the polynomial ring in one indeterminate over R) then R[x]/I is Jacobson radical if and only if R/I ′ [x] is Jacobson radical, where I ′ is the ideal of R generated by coefficients of polynomials from I. Also if R is a nil ring and I is a primitive ideal of R[x] then I = M [x] for some ideal M of R. It was asked by Beidar, Fong and Puczy lowski [1] whether polynomial rings over nil rings are not (right) primitive. We show that affirmative answer to this question is equivalent to the Köthe conjecture. We also answer in the negative Question 2 from [1] (Corollary 1). It is known that if a polynomial ring R[x] is primitive then R need not be primitive [3] (see also Bergman's example in [5]). Let R be a prime ring and I a nonzero ideal of R. Then R is a primitive ring if and only if I is a primitive ring [6]. Since the Hodges example has a nonzero Jacobson radical it follows that polynomial rings over Jacobson radical rings can be right and left primitive (see also Theorem 3). We recall some definitions after [9] (see also [2], [5]). A right ideal of a ring R is called modular in R if and only if there exists an element b ∈ R such that a − ba ∈ Q for every a ∈ R. If Q is a modular maximal right ideal of R then for every r / ∈ Q, rR + Q = R. An ideal P of a ring R is right primitive in R if and only if there exists a modular maximal right ideal Q of R such that P is the maximal ideal contained in Q. In this paper R[x] denote the polynomial ring in one indeterminate over R. Given polynomial g ∈ R[x] by deg(g) we denote the degree of R, i.e., the
Communications in Algebra, Dec 31, 2009
Let K be a field and let A be a finitely generated prime K-algebra. We generalize a result of Smi... more Let K be a field and let A be a finitely generated prime K-algebra. We generalize a result of Smith and Zhang, showing that if A is not PI and does not have a locally nilpotent ideal, then the extended centre of A has transcendence degree at most GKdim(A) − 2 over K. As a consequence, we are able to show that if A is a prime K-algebra of quadratic growth, then either the extended centre is a finite extension of K or A is PI. Finally, we give an example of a finitely generated non-PI prime K-algebra of GK dimension 2 with a locally nilpotent ideal such that the extended centre has infinite transcendence degree over K. * The first author thanks NSERC for its generous support.
Advances in Mathematics, Dec 1, 2009
It is proved that over every countable field K there is a nil algebra R such that the algebra obt... more 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].
arXiv (Cornell University), May 9, 2005
We construct a nil algebra over a countable field which has finite but non-zero Gelfand-Kirillov ... more We construct a nil algebra over a countable field which has finite but non-zero Gelfand-Kirillov dimension.
arXiv (Cornell University), Apr 11, 2018
This paper introduces the notion of a strongly prime ideal, and shows that the largest solvable i... more This paper introduces the notion of a strongly prime ideal, and shows that the largest solvable ideal in a finite brace equals the intersection of all strongly prime ideals in this brace. This is used to generalise some well known results from ring theory into the context of braces and pre-Lie algebras. Several open questions are also posed.
arXiv (Cornell University), Apr 6, 2015
A well-known theorem by S.A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] ... more 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.
Israel Journal of Mathematics, May 29, 2012
For an arbitrary countable field, we construct an associative algebra that is graded, generated b... more For an arbitrary countable field, we construct an associative algebra that is graded, generated by finitely many degree-1 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.
arXiv (Cornell University), May 19, 2017
Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerat... more Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf-Galois extensions. These connections and some of their consequences are explored in this paper. We produce several new families of solutions related in many different ways with rings, near-rings and groups. We also study the solutions of the YBE that skew braces naturally produce. We prove, for example, that the order of the canonical solution associated with a finite skew brace is even: it is two times the exponent of the additive group modulo its center.
arXiv (Cornell University), Nov 17, 2010
We show that if k is a countable field, then there exists a finitely generated, infinite-dimensio... more 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.
Asian-european Journal of Mathematics, Jun 1, 2009
Let K be a field of infinite transcendence degree and let A be a finitely generated K-algebra. Su... more 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.
Journal of Algebra, May 1, 2013
It is shown that Golod-Shaferevich algebras of a reduced number of defining relations contain non... more It is shown that Golod-Shaferevich 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.
Journal of Pure and Applied Algebra, Jun 1, 2007
It is shown that for every countable field K , there is a finitely generated graded Jacobson radi... more It is shown that for every countable field K , there is a finitely generated graded Jacobson radical algebra over K of Gelfand-Kirillov dimension two. Examples of finitely generated Jacobson radical algebras of Gelfand-Kirillov dimension two over algebraic extensions of finite fields of characteristic 2 were earlier constructed by Bartholdi [L. Bartholdi, Branch Rings, thinned rings, tree enveloping rings, Israel J. Math. (in press)].
arXiv (Cornell University), Aug 4, 2022
Let A be a brace of cardinality p n where p > n + 1 is prime, and ann(p i) be the set of elements... more Let A be a brace of cardinality p n where p > n + 1 is prime, and ann(p i) be the set of elements of additive order at most p i in this brace. A pre-Lie ring related to the brace A/ann(p 2) was constructed in [8]. We show that there is a formula dependent only on the additive group of the brace A which reverses the construction from [8]. As an application example it is shown that the brace A/ann(p 4) is the group of flows of a left nilpotent pre-Lie algebra.
arXiv (Cornell University), Sep 1, 2015
It is shown that over an arbitrary field there exists a nil algebra R whose adjoint group R o is ... more 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 [37], Rump introduced braces and radical chains A n+1 = A • A n and A (n+1) = A (n) • 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 is a finite brace whose adjoint group A o is nilpotent 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 non-degenerate involutive set-theoretic Yang-Baxter equation. It is worth noticing that by a result by Gateva-Ivanova [17] braces are in oneto-one correspondence with braided groups with involutive braided operators.
arXiv: Rings and Algebras, 2015
We show that there exists a nil ring R whose adjoint group $R^{o}$ is not an Engel group. This an... more We show that there exists a nil ring R whose adjoint group $R^{o}$ is not an Engel group. This answers a question by Amberg and Sysak from 1997 [5], and also answers related questions from [3, 30]. We also show that over an arbitrary uncountable field there is a nil algebra R whose adjoint group $R^{o}$ is not an Engel group. This answers a recent question by Zelmanov. In [25], Rump showed some surprising connections between Jacobson radical rings and solutions to the Young-Baxter equation, and proved that every Jacobson radical ring yields a solution to the Young-Baxter equation. In particular, he showed that Jacobson radical rings are in one-to-one correspondence with two-sided braces. In this context, Cedo, Jespers and Okninski [13, 12] asked which groups are multiplicative groups of braces. A similar question in the language of ring theory was asked in [4, 5]. We obtain some results related to these questions in Corollaries 3 and 4. It is also worth noticing that, by a result by...
European Congress of Mathematics Amsterdam, 14–18 July, 2008
Journal of the American Mathematical Society, 2007
The famous 1960’s construction of Golod and Shafarevich yields infinite dimensional nil, but not ... more 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.
Serdica. Mathematical Journal, 2001
The Köthe conjecture states that if a ring R has no nonzero nil ideals then R has no nonzero nil ... more The Köthe conjecture states that if a ring R has no nonzero nil ideals then R has no nonzero nil one-sided ideals. Although for more than 70 years significant progress has been made, it is still open in general. In this paper we survey some results related to the Köthe conjecture as well as some equivalent problems.
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Papers by Agata Smoktunowicz