Faculty of Science Departement of Mathematics Nil and Jacobson radicals in semigroup graded rings Master thesis submitted in partial fulfillment of the requirements for the degree of Master in Mathematics Carmen Mazijn Promotor: Prof. Dr. E. Jespers AUGUST 2015 Acknowledgements When we started our last year of the Master in Mathematics at VUB, none of us knew how many hours we would spend on the reading, understanding and writing of our thesis. This final product as conclusion of the master was at that point only an idea. The subject was chosen, the first papers were read and the first words were written down. And more words were written, more books were consulted, more questions were asked to our promoters. Writing a Master thesis is a journey. Even though next week everyone will have handed in there thesis, we don’t yet understand clearly where this journey took us, for the future is unknown. First of all I would like to thank professor Eric Jespers, for giving me the chance to grow as mathematician in the past years. With every semester the interest in Algebra and accuracy as mathematician grew. Thank you for the guidance through all the books and papers to make this a consistent dissertation. Secondly I would like to thank all my classmates and compa˜nerosde clase. For frowned faces when we didn’t get something in class, the laughter when we realized it was a ctually quite trivial or sometimes not even at all. For the late night calls and the interesting discussions. It was a pleasure. Tambi´engracias a todos que conoc´ı en Granada en mi a˜node Erasmus. Era mucho mas que un a˜node erasmus y sois mucho mas que mis amigos. Hasta pronto. Lastly, but probably mostly, I would like to thank my parents and little brother. For actually listening most of the times when I tried to explain something. For the constant support during these year. For making me come home now and then, as Bruges will always be one of my homes. Carmen Mazijn Brussels, August 2015 i Samenvatting In het kader van deze masterproef werken we in niet-commutatieve ring theorie, en meer specifiek in radicaal theorie van gegradeerde ringen. In het eerste hoofdstuk worden radicalen en gegradeerde ringen gedefinieerd. Er worden respectievelijk voorbeelden gegeven van verscheidene radicalen zoals het nil, Jacobson, priem en Brown-McCoy radicaal alsook van gradaties op verschillende rin- gen. Het is nuttig om deze radicalen te bestuderen omdat het kan helpen bij het beantwoorden van het Koethe vermoeden. Het Koethe vermoden stelt dat een ring die geen niet nulle tweezijdige nil idealen heeft, ook geen niet nulle eenzijdige nil ide- alen heeft. Voor het Koethe vermoeden bestaan er equivalente stellingen die gebruik maken van het nil radicaal of het Jacobson radicaal. In het tweede hoofdstuk wordt een chronologisch overzicht gegeven van relevante stellingen en vragen in dit onderzoeksgebied. In het derde hoofdstuk wordt er bewezen dat het Jacobson en nil radicaal in een Z-gegradeerde ring homogeen zijn. Dit zijn respectievelijk resultaten van Bergman in 1975 en Smoktunowicz in 2006. Verder bewees Smoktunowicz dat de homogene deelringen van een Jacobson radicaal ring Jacobson radicaal zijn. Men kan zich afvragen voor welke semigroepgradaties de stellingen van Smok- tunowicz waar blijven. Dit is het uitgangspunt van hoofdstuk vier. Mazurek, Nielsen en Ziembowski gaven recent antwoord op deze vragen. Als de semigroep S abels, annuleerbaar en torsie-vrij is, dan is het nil radicaal van een S-gegradeerde ring ho- mogeen. Verder is een nodige voorwaarde voor de semigroep S opdat de homogene deelringen van een Jacobson radicaal S-gegradeerde ring Jacobson radicaal zou zijn, dat \ T n = ∅ n≥1 voor alle eindig voortgebrachte deelsemigroepen T van S. Een voorbeeld zal tonen dat dit geen voldoende voorwaarde is. Als slot worden worden in het laatste hoofdstuk nog enkele open vragen opgelijst. ii Contents Acknowledgements i Samenvatting ii Introduction 1 1 Background 3 1.1 Rings and modules . 3 1.2 Introduction to radical theory . 4 1.2.1 Definitions . 4 1.2.2 Examples of radicals . 6 1.2.3 Examples of radical constructions . 9 1.2.4 The insignificance of the multiplicative unit in radicals . 12 1.3 Graded rings . 13 1.3.1 Definitions . 13 1.3.2 Graded ideals . 15 1.3.3 Examples . 16 1.4 The radical of a graded ring . 18 2 History 20 2.1 A radical chain . 20 2.2 Historical overview . 21 3 Radicals of Z-graded rings 27 3.1 Introduction . 27 3.2 Amitsur’s result on the Jacobson radical of R[X] . 27 3.3 Bergman’s results . 31 3.3.1 Simple modules and finite extensions . 31 3.3.2 The Jacobson radical of Z-graded rings . 32 3.3.3 The Jacobson radical of rings graded by a torsion-free abelian group . 34 iii 3.4 Smoktunowicz’s results . 35 3.4.1 The prime radical of Z-graded rings . 35 3.4.2 The nil radical of Z-graded rings . 36 3.4.3 Jacobson radical rings that are Z-graded . 39 3.4.4 Graded-nil rings and the Brown-McCoy radical . 40 4 Radicals in semigroup graded rings 41 4.1 The nil radical in semigroup graded rings . 41 4.1.1 Ideal functions . 42 4.1.2 Cancellative semigroups . 43 4.1.3 Torsion-free semigroups . 44 4.1.4 Abelian, cancellative, torsion-free semigroups . 44 4.2 Homogeneous Jacobson radical subrings of Jacobson radical rings . 45 4.2.1 A power series construction . 46 4.2.2 A large class of examples . 47 4.2.3 More examples . 47 5 Open questions 49 Bibliography 50 iv Introduction The scope of this thesis takes places in non-commutative ring theory. One of the important open problems is whether the Koethe conjecture is true or false. Conjecture. A ring R with no nonzero nil (two-sided) ideals does not have a nonzero nil one-sided ideal. Since 1930 mathematicians have tried to find a solution to this question up and till today without result. A proven method of tackling difficult questions is to find equivalent statements. As the Koethe conjecture talks about nil ideals, there exist equivalent statements concerning nil rings, the nilradical and also the Jacobson rad- ical. An other tactic to handle problems is by splitting it up into smaller parts or dealing with specific but important cases. For example, consider rings with some inherent structure such as being a graded ring. The main question of this thesis is to determine when radicals in graded rings are homogeneous. We focus on the nil and Jacobson radical as these are the most know and important ones, but we will also deal with examples of other radicals. The main emphasis is on Z-graded rings, but some results will be stated in a more gen- eral context as torsion-free abelian group graded and torsion-free cancellative abelian semigroup graded rings. The first chapter starts with some background on the two main concepts men- tioned in the title of this thesis, namely radicals and graded rings. First we give the definition of a radical class and consider the nil, Jacobson, prime and Brown-McCoy radical. Next we give the definition of graded rings and graded ideals. We consider examples of natural gradings but also of some more creative gradings. The second chapter gives an overview of the field of radicals since Koethe postu- lated his conjecture in 1930. Theorems, equivalent statements and more questions related to the Koethe conjecture are put in chronological order. In the third chapter the nil and Jacobson radical in Z-graded rings are studied. In 1965 Amitsur showed that the Jacobson radical of the polynomial ring R[X] is 1 homogeneous. Bergman showed in 1975 that the Jacobson radical of every ring graded by Z, and more general graded by a torsion-free abelian group, is homogeneous. It then took 30 years to be proven that the same holds for the nil radical. This was shown by Smoktunowicz in 2006. She also proved that for some Jacobson radical Z-graded rings each subring that is generated by homogeneous elements is Jacobson radical. We will give complete proves for these results. It is a natural question to investigate for which graded rings (by an arbitrary semigroup) the above results remain valid. In 2006 Smoktunowicz posted two concrete questions in this context. This is the topic of chapter four. First we will show a recent result of Mazurek, Nielsen and Ziembowski stating that the nil radical of a ring graded by a torsion-free, cancellative, abelian semigroup is homogeneous. They also investigated when Jacobson radical semigroup graded rings are such that all subrings are Jacobson radical. A sufficient but not necessary condition is that \ T n = ∅ n≥1 for every finitely generated subsemigroup T . An example will show that this is not a necessary condition. Complete proofs will be given of the results. In the final chapter we finish by stating some open questions concerning related problems. 2 Chapter 1 Background The scope of this work takes place in ring theory. After defining what is meant by rings and modules, we will define the two main concepts that are mentioned in the title, i.e. radicals and graded rings. We start with a short introduction to radical theory. We continue with defining and giving examples of graded rings. Afterwards we consider the definition of a graded radical. 1.1 Rings and modules An associative ring (R, +, ·) is a set R with two binary operations, the addition + and the multiplication ·, such that (R, +, 0) is an abelian group with a neutral element 0, (R, ·) is a semigroup, such that the following conditions (the distributivity laws) are satisfied for all r, s, t ∈ R: r(s + t) = rs + rt and (s + t)r = sr + tr.