FREE ELECTRONIC LIBRARY - Dissertations, online materials

Pages:   || 2 |

«Abstract. In this paper we survey some results on the structure of noncommutative rings. We focus particularly on nil rings, Jacobson radical rings ...»

-- [ Page 1 ] --

Some results in noncommutative ring theory

Agata Smoktunowicz

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


Mathematics Subject Classification (2000). 16-02, 16-06, 16N40, 16N20, 16N60, 16D60,


Keywords. Nil rings, Jacobson radical, algebraic algebras, prime algebras, growth of algebras,

the Gelfand–Kirillov dimension.

1. Introduction We present here a brief outline of results and examples related mainly to noncommu- tative nil rings. In this exposition rings are noncommutative and associative. A vector space R is called an algebra (or a K-algebra) if R is equipped with a binary operation ∗ : (R, R) → R, called multiplication, such that for any a, b, c ∈ R and for any α ∈ K, we have (a + b) ∗ c = a ∗ c + b ∗ c, a ∗ (b + c) = a ∗ b + a ∗ c, (a ∗ b) ∗ c = a ∗ (b ∗ c), α(a ∗ b) = (αa) ∗ b = a ∗ (αb).

It is known that simple artinian rings, commutative simple rings and simple right noetherian rings of characteristic zero have unity elements [35]. In this text, rings are usually without 1. In fact nil rings and Jacobson radical rings cannot have unity elements.

2. Nil rings The most important question in this area is the Köthe Conjecture, first posed in 1930.

Köthe conjectured that a ring R with no nonzero nil (two-sided) ideals would also have no nonzero nil one-sided ideals, [24], see also [15] and [27]. This conjecture is still open despite the attention of many noncommutative algebraists. It is a basic question concerning the structure of rings.

Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society 260 Agata Smoktunowicz The truth of the conjecture has been established for many classes of rings: typically, one proves that for a given class of rings, the sum of all nil one-sided ideals is nil. The most famous examples of such results are the proof of the conjecture in the case of algebras over uncountable fields by Amitsur, and the fact that nil ideals are nilpotent in the class of noetherian rings, proved by Levitski, see [27]. However, as indicated above, Köthe’s conjecture is still open in the general case.

An element r in a ring R is said to be nilpotent if r n = 0 for some n. A ring R is a nil ring if every element of R is nilpotent, and the ring R is nilpotent if R n = 0 for some n. A more appropriate definition in the case of infinitely generated rings is the following. A ring R is locally nilpotent if every finitely generated subring of R is nilpotent. A thoroughunderstanding of nil and nilpotent rings is important for an attempt to understanding general rings.

In addition, nil rings have some applications in group theory. The following famous theorem was proved in 1964 by Golod and Shafarevich. For every field F there exists a finitely generated nil F -algebra R which is not nilpotent ([20]). Recall that a group G is said to be torsion (or periodic) if every g ∈ G has a finite order. Golod used the group 1 + R, when F has positive characteristic, to get a counterexample to the General Burnside Problem: Let G be a finitely generated torsion group. Is G necessarily finite?

There are many open questions concerning nil rings. As mentioned before, the most important is now known as the Köthe Conjecture and was posed by Köthe in 1930: if a ring R has no nonzero nil ideals, does it follow that R has no nonzero nil one-sided ideals? Köthe himself conjectured that the answer would be in the affirmative ([24], [27], [37]).

There are many assertions equivalent to the Köthe Conjecture: For example, the

following are equivalent to Köthe’s conjecture:

1. The sum of two right nil ideals in any ring is nil.

2. (Krempa [26]) For every nil ring R the ring of 2 by 2 matrices over R is nil.

3. (Fisher, Krempa [18]) For every ring R, R G is nil implies R is nil (G is the group of automorphisms of R, R G the set of G-fixed elements).

4. (Ferrero, Puczylowski [17]) Every ring which is a sum of a nilpotent subring and a nil subring must be nil.

5. (Krempa [26]) For every nil ring R the polynomial ring R[x] in one indeterminate over R is Jacobson radical.

6. (Smoktunowicz [44]) For every nil ring R the polynomial ring R[x] in one indeterminate over R is not left primitive.

–  –  –

Recall that a ring R is Jacobson radical if for every r ∈ R there is r ∈ R such that r + r + rr = 0. Every nil ring is Jacobson radical. The largest ideal in a ring R, which is Jacobson radical is called the Jacobson radical of R. The Jacobson radical of a ring R equals the intersection of all (right) primitive ideals of R (I is a primitive ideal in R if I /R is primitive). Recall that a ring R is (right) primitive if there is a maximal right ideal Q such that Q + I = R for every nonzero ideal I in R and there is b ∈ R such that br − r ∈ Q for every r ∈ R ([13]).

The Köthe Conjecture is said to hold for a ring R if the ideal generated by the nil left ideals of R is nil. Köthe’s conjecture holds for the class of Noetherian rings (Levitzki, [27], [32]), Goldie rings (Levitzki, [32]), rings with right Krull dimension (Lenagan [29], [15]), monomial algebras (Beidar, Fong [6]), PI rings (Rasmyslow– Kemer–Braun [14], [34], [22], [12] ), algebras over uncountable fields (Amitsur [27], [36]).

There are many related results, some are indicated in the following.

Theorem 2.1 (Levitzki; [32]).

Let R be a right Noetherian ring. Then every nil one-sided ideal of R is nilpotent.

Theorem 2.2 (Lenagan [29]).

If R has right Krull dimension, then nil subrings of R are nilpotent.

Theorem 2.3 (Gordon, Lenagan and Robson, Gordon and Robson; [15]).

If R has right Krull dimension, then the prime radical of R is nilpotent.

The prime radical of R is a nil ideal and is equal to the intersection of all prime ideals in R.

Theorem 2.4 (Beidar, Fong [6]).

Let X be a nonempty set, Z = X the free monoid on X, Y an ideal of the monoid Z, and F a field. Then the Jacobson radical of the monomial algebra F [Z/Y ] is locally nilpotent.

In the case of characteristic zero the result is due to Jaspers and Puczylowski, [21]. Earlier, Belov and Gateva-Ivanova [10] showed that the Jacobson radical of a finitely generated monomial algebra over a field is nil. However, it is not true that the Jacobson radical of a finitely generated monomial algebra is nilpotent, since it was shown by Zelmanov [50] that there is a finitely generated prime monomial algebra with a nonzero locally nilpotent ideal.

Theorem 2.5 (Razmyslov–Kemer–Braun [34], [22], [12]; [14]).

If R is a finitely generated PI-algebra over a field then the Jacobson radical of R is nilpotent.

Razmyslov [34] proved this for rings satisfying all identities of matrices, Kemer [22] for algebras over fields of characteristic zero. Later Braun [12] proved the nilpotency of the radical in any finitely generated PI algebra over a commutative noetherian ring. Amitsur has previously shown that the Jacobson radical of a finitely generated PI algebra over a field is nil.

Another famous result is the Nagata–Higman Theorem:

262 Agata Smoktunowicz Theorem 2.6 (Nagata–Higman; [19]). If A is an associative algebra of characteristic p such that a n = 0 for all a ∈ A and p n or p = 0 then A is nilpotent.

For interesting results related to Nagata–Higman’s theorem see [19].

A theorem of Klein [23] asserts that if R is a nil ring of bounded index then R[x] is a nil ring of bounded index.

In 1956 Amitsur [27] showed that if R is a nil algebra over an uncountable field, then the polynomial ring R[x] in one indeterminate over R is also nil. The situation is completely different for countable fields, as was shown by the author in 2000.

Theorem 2.7 (Smoktunowicz [43]).

For every countable field K there is a nil Kalgebra N such that the polynomial ring in one indeterminate over N is not nil.

This answers a question of Amitsur. Another important theorem by Amitsur is the following.

Theorem 2.8 (Amitsur; [27]).

Let R be a ring. Then the Jacobson radical of the polynomial ring R[x] is equal to N[x] for some nil ideal N of R.

In 1956 Amitsur conjectured that if R is a ring, and R[x] has no nil ideals then it is semiprimitive (i.e. the Jacobson radical of R[x] is zero). This assertion is true for many important classes of rings, as mentioned above. However, the following theorem shows that this conjecture does not hold in general: There is a nil ring N such that the polynomial ring in one indeterminate over N is Jacobson radical but not nil ([41]). For some generalizations of this theorem see [45]. This theorem is true in a more general setting: For every natural number n, there is a nil ring N such that the polynomial ring in n commuting indeterminates over N is Jacobson radical but not nil.

Recall that, as shown by Krempa in [26], Köthe’s conjecture is equivalent to the assertion that polynomial rings over nil rings are Jacobson radical. However, homomorphic images of polynomial rings over nil rings with nonzero kernels are often Jacobson radical, as is shown by the next result.

Theorem 2.9 (Smoktunowicz [44]).

Let R be a nil ring and R[x] the polynomial ring in one indeterminate over R. Let I be an ideal in R[x] and M the ideal of R generated by coefficients of polynomials from I. Then R[x]/I is Jacobson radical if and only if R[x]/M[x] is Jacobson radical.

The following are interesting open questions on nil rings.

Question 1 (Latyshev, [16], pp. 12). Let A be an associative algebra with a finite number of generators and relations. If A is a nil algebra must it be nilpotent?

–  –  –

3. Algebraic algebras The most well-known question in this area is the Kurosh Problem ([15], [36]). Let R be a finitely generated algebra over a field F such that R is algebraic over F. Is R finite dimensional over F ?

This problem has a negative solution in general. The famous construction of Golod and Shafarevich in the 1960s produced a finitely generated nil algebra which is not nilpotent ([20]). This was then used to construct a counterexample to the Burnside Conjecture, one of the biggest outstanding problems in group theory at that time. Zelmanov was later awarded the Fields Medal for his solution of the Restricted Burnside Problem [27].

However, the Kurosh Problem is still open for the key special case of a division


Question 3 (Kurosh’s problem for division rings [16], [36]). Let R be a finitely generated algebra over a field F such that R is algebraic over F and R is a division ring. Does it follow that R a finite dimensional vector space over its center?

Again, as with the nil ring problems, there are many partial results. The Kurosh Problem for division rings is still open in general, but it is answered affirmatively for F finite and for F having only finite algebraic field extensions, in particular, for F algebraically closed ([36]). By Levitzki’s and Kaplanski’s theorem, Kurosh’s conjecture is also true if there is a bound on the degree of elements in R ([15]).

It is unknown whether Kurosh’s problem for division rings has a positive answer in the case of algebras over uncountable fields. Also the following question is still open: Is Kurosh’s conjecture true for division rings with finite Gelfand–Kirillov dimension, and in particular for division rings with quadratic growth? There are obvious 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 graded nil, i.e., all homogeneous elements are nil. On the other hand, the Kurosh Problem has a negative solution for rings with finite Gelfand–Kirillov dimension ([30]), for simple rings ([42]), for primitive rings ([2]), for finitely generated primitive rings ([8]), and for finitely generated algebraic primitive rings ([9]). However, a natural question

arising from the general Kurosh Problem remains open:

Question 4 (Small’s question). Let R be a finitely generated simple algebra with 1 over a field F such that R is algebraic over F. Is R a finite dimensional vector space over its center?

Another open question on division rings, which has been around for years, is the


Question 5. Let K be a field and let R be a finitely generated algebra which is a division ring. Does it follow that R is a finitely generated vector space over K?

264 Agata Smoktunowicz As far as I know this question is very much open even with various conditions, like e.g. Gelfand–Kirillov dimension 2. It has been shown by Small ([38]) that a division ring which is a homomorphic image of a graded noetherian ring (of course, by a non graded ideal) must be finite dimensional. There is a similar open question

concerning rings:

Question 6 ([16], p. 20). Does there exist an infinite associative division ring which is finitely generated as a ring?

Pages:   || 2 |

Similar works:

«PERSISTENT HYPERPLASTIC PRIMARY VITREOUS: DIAGNOSIS, TREATMENT AND RESULTS* BY Zane F Pollard, MD INTRODUCTION Persistent hyperplastic primary vitreous, known as PHPV, poses a challenge to the ophthalmologist because of the difficulty in obtaining a good visual result from treatment. Following posterior lenticonus, PHPV is the second most common cause of acquired cataract during the first year of life. In some cases, a cataract associated with PHPV may be present at birth. The purpose of this...»

«XIII CONGRESO INTERNACIONAL SOBRE PATRIMONIO GEOLÓGICO Y MINERO. Manresa2012, C.41 p. 393400. ISBN nº 978 – 99920 – 1 – 769 2 THE GEOLOGICAL AND PALEONTOLOGICAL HERITAGE OF MANRESA MUNICIPALITY (CATALONIA, SPAIN) Oriol OMS1, Ferran CLIMENT2,3, David PARCERISA4, Josep Maria MATA-PERELLÓ4, Joan POCH1,2 1 Universitat Autònoma de Barcelona. Campus Bellaterra. Departament de Geologia, Facultat de Ciències. 08193 Cerdanyola del Vallès (Spain), joseporiol.oms@cat 2 GEOSEI...»

«10.1177/1523422305274528 Advances in Developing Human Resources May 2005 McLean / ORGANIZATIONAL CUL TURE’S INFLUENCE Organizational Culture’s Influence on Creativity and Innovation: A Review of the Literature and Implications for Human Resource Development Laird D. McLean The problem and the solution. The majority of the literature on creativity has focused on the individual, yet the social environment can influence both the level and frequency of creative behavior. This article reviews...»

«IN THE UNITED STATES DISTRICT COURT FOR THE EASTERN DISTRICT OF PENNSYLVANIA DATA COMM COMMUNICATIONS, INC., : CIVIL ACTION ERIC J. PERRY, and : LOUIS SILVER : :v. : : MARVIN WALDMAN, : HENRIETTA ALBAN, : THE REMINGTON GROUP, and : ANDREW BOGDANOFF : NO. 97-0735 MEMORANDUM AND ORDER HUTTON, J. July 15, 1999 Presently before the Court are the Motion for Summary Judgment of Defendants Marvin Waldman and Henriette Alban (collectively, “Defendants”) (Docket No. 72), the Plaintiffs’ Response...»

«Public Disclosure Authorized 76654 Competition and Scope of Activities in Financial Services Public Disclosure Authorized Stijn Claessens • Daniela Klingebiel This article analyzes the costs and benefits of different degrees of competition and different configurations of permissible activities in the financial sector and discusses the related implications for regulation and supervision. Theory and experience demonstrate the importance of competition for efficiency and confirm that a...»

«INTRODUCTION COMING TO TERMS WITH FILM NOIR AND EXISTENTIALISM “Let’s get the details fixed first.”1 —Sam Spade to Caspar Gutman in the film The Maltese Falcon (1941) The alleged official arrival of existentialism to the United States was marked by the much publicized visit of French existentialists, Jean-Paul Sartre, Albert Camus, Simone de Beauvoir, and Maurice Merleau-Ponty to New York soon after World War II. Newspapers and popular magazines, aware of the American fascination...»

«Trespassed on Press 177 Journalists Subjected of Press Freedom Violation in 177 Days of Madhesh Movement, August 16, 2015 – February 12, 2016 Binod Dhungel May 3, 2016 World Press Freedom Day Foreword 5 Introduction 6 General Background 6 Methodology 7 Mounted Aggression Against Press 8 1. After the Promulgation of the Constitution 8 Attacks on media houses 8 Journalists under bomb and bullet 8 Media persons manhandled/misbehaved 9 Journalists threatened 10...»

«VILLA DES BIJOUX by MATTHEW ASPREY for J. G. THIS IS A STORY of discovery and also a story of loss. I’ll begin in Paris, in the wintertime, in the lobby of the Hotel Bolovens in St-Germain-des-Prés. I was sitting with my fat half-sister Alexis on a sofa. I was wearing a horizontallystriped blue-grey pullover and a red foulard with white polka-dots. I looked fantastic. Alexis, unimaginatively practical, hid her figure in a black woollen overcoat. “Leon will help us,” I reassured her. 1...»

«Properties of Proteins Experiment #8 To study chemical and physical properties of proteins from natural sources (egg Objective: and milk) and some chemical reactions of amino acid residues in these proteins, as well as the effects of denaturing agents on these proteins. Introduction Proteins are polymers of the 20 common amino acids. The amino acids are linked through their -amino and -carboxyl groups to form peptide bonds. The proteins you will study are casein (milk protein) and albumin (in...»

«CHAPTER INFORMATIONAL MANUAL INTRODUCTION The purpose of this manual is threefold: 1. To assist the Chapter Officers, especially the Secretary, in defining the required procedures necessary to manage their Chapter in a professional and efficient manner.2. To urge the chapter to increase the visibility of their chapter activity by sending regular press releases to your local newspapers and regional Greek-American Press. 3. A guideline as to how to increase your chapter’s membership and more...»

«Third UN World Conference on Disaster Risk Reduction, 2015, Sendai, Japan Public Forum Side Event Mainstreaming Ecosystem-based Disaster Risk Reduction and Reconstruction: Lessons Learned from 3/11 for the World and the Future 14 March 2015, 13:00–16:00 TKP Garden City Sendai Kotodai Report United Nations University Institute for the Advanced Study of Sustainability June 2015 On 14 March 2015, the international symposium ‘Mainstreaming Ecosystem-based Disaster Risk Reduction and...»

«Teaching Gender in Social Work Teaching with Gender. European Women’s Studies in International and Interdisciplinary Classrooms. A book series by ATHENA Edited by Vesna Leskošek Edited by Vesna Leskošek Teaching Gender in Social Work Teaching with Gender. European Women’s Studies in International and Interdisciplinary Classrooms. A book series by ATHENA © Alice Salomon Archive der ASFH Berlin. Welfare Archives of the Private Charity Organisation Society in Berlin: Students studying the...»

<<  HOME   |    CONTACTS
2016 www.dissertation.xlibx.info - Dissertations, online materials

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.