CLEAN RINGS & CLEAN GROUP RINGS Nicholas A. Immormino A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2013 Committee: Warren Wm. McGovern, Advisor Rieuwert J. Blok, Advisor Sheila J. Roberts, Graduate Faculty Representative Mihai D. Staic ii ABSTRACT Warren Wm. McGovern, Advisor Rieuwert J. Blok, Advisor A ring is said to be clean if each element in the ring can be written as the sum of a unit and an idempotent of the ring. More generally, an element in a ring is said to be clean if it can be written as the sum of a unit and an idempotent of the ring. The notion of a clean ring was introduced by Nicholson in his 1977 study of lifting idempotents and exchange rings, and these rings have since been studied by many different authors. In our study of clean rings, we classify the rings that consist entirely of units, idempotents, and quasiregular elements. It is well known that the units, idempotents, and quasiregular elements of any ring are clean. Therefore any ring that consists entirely of these types of elements is clean. We prove that a ring consists entirely of units, idempotents, and quasiregular elements if and only if it is a boolean ring, a local ring, isomorphic to the direct product of two division rings, isomorphic to the full matrix ring M2(D) for some division ring D, or isomorphic to the ring of a Morita context with zero pairings where both of the underlying rings are division rings. We also classify the rings that consist entirely of units, idempotents, and nilpotent elements. In our study of clean group rings, we show exactly when the group ring Z(p)Cn is clean, where Z(p) is the localization of the integers at p, and Cn is the cyclic group of order n. It is well known that the group ring Z(7)C3 is not clean even though the group ring Z(p)C3 is quasiclean, semiclean, and \Sigma -clean for any prime p, and 2-clean for any prime p 6= 2. We prove that Z(p)C3 is clean if and only if p \not \equiv 1 modulo 3. More generally, we prove that the group ring Z(p)Cn is clean if and only if p is a primitive root of m, where n = pkm and p does not divide m. We also consider the problems of classifying the groups G whose group rings RG are clean for any clean ring R, and of classifying the rings R such that the group ring RG of any locally finite group G over the ring R is clean. iii ACKNOWLEDGMENTS I would like to thank all of the wonderful mathematics teachers that I had the opportunity to work with and learn from since I first became enamored with mathematics in the fourth grade. In particular, I would like to thank Thomas W. Hungerford, Curtis D. Bennett, Corneliu G. Hoffman, Sergey Shpectorov, and Warren Wm. McGovern for teaching me about ring theory. I would like to thank Rieuwert J. Blok, Mihai D. Staic, and Sheila J. Roberts for their service on my dissertation committee, and I would like to thank Warren Wm. McGovern for being my advisor. I would also like to thank Tong Sun and Marcia Seubert for all their help, especially over the past year. iv LIST OF SYMBOLS Z the integers Q the rational numbers Z=nZ the integers modulo n Z(n) the rational numbers with denominators relatively prime to n Zp the p-adic integers \phi the Euler phi function ordn(a) the multiplicative order of a modulo n A \times B the direct product of A and B Eij the ijth matrix unit Mn(R) the set of n \times n matrices with entries from R Tn(R) the set of n \times n upper triangular matrices with entries from R R[X] the set of polynomials with coefficients in R Cn the cyclic group of order n Sn the symmetric group on n elements Cp1 the Pr\"ufer p-group hgi the cyclic group generated by g Gk the direct product of k copies of the group G Gp the p-primary component of the group G U(R) the set of all units of the ring R ID(R) the set of all idempotents of the ring R QR(R) the set of all quasiregular elements of the ring R J(R) the Jacobson radical of the ring R Fq the finite field of order q char K the characteristic of the field K \Phin the nth cyclotomic polynomial over a field T (R; M) the trivial extension of R by M (A; B; M; N; ; ') the Morita context of A; B; M; N; ; and ' RG the group ring of G over R \Delta the augmentation ideal of a groupring v TABLE OF CONTENTS CHAPTER ONE: Clean Rings 1 x1 Some Properties of Clean Rings...................................... 1 Basic Properties............................................. 1 Lifting Idempotents........................................... 2 Two More Properties.......................................... 4 Some Notes................................................ 6 x2 Some Examples of Clean Rings...................................... 8 Division, Boolean & Local Rings.................................. 8 Perfect & Semiperfect Rings..................................... 10 Regular & \pi -Regular Rings...................................... 14 Some Notes................................................ 17 x3 An Interesting Subclass of Clean Rings................................. 20 Abelian Rings............................................... 20 Nonabelian Rings............................................ 24 A Special Case.............................................. 29 Conclusions................................................ 32 Some Notes................................................ 33 CHAPTER TWO: Clean Group Rings 36 x4 Preliminaries.................................................. 36 Some Results on Semiperfect Group Rings............................ 36 Some Results on Clean Group Rings................................ 38 Some Notes................................................ 41 x5 Clean Group Rings, I............................................. 43 The Group Ring Z(p)Cn ........................................ 43 vi The Group Ring Z(p)G ......................................... 46 Two More Results............................................ 47 Some Notes................................................ 48 x6 Clean Group Rings, II............................................ 50 Abelian Groups.............................................. 51 Nonabelian Groups........................................... 55 x7 Clean Group Rings, III............................................ 56 Three Examples............................................. 56 Some Notes................................................ 58 REFERENCES 60 INDEX 67 1 CHAPTER ONE Clean Rings A ring is said to be clean if every element in the ring can be written as the sum of a unit and an idempotent of the ring. More generally, an element in a ring is called clean if it can be written as the sum of a unit and an idempotent of the ring. These rings were introduced by Nicholson [52] in his study of lifting idempotents and exchange rings. In this chapter we review some well-known properties and examples of clean rings, and then we classify the rings that belong to an interesting subclass of clean rings. For a more general introduction to clean rings, see the survey of clean rings given by Nicholson and Zhou [55]. For a different perspective on this class of rings, see the history of commutative clean rings given by McGovern [47]. In what follows, all rings are associative with identity 1 6= 0, all modules are unitary, and we generally use the notation and terminology of [41] and [42], which we list as our main references on noncommutative rings. x1 Some Properties of Clean Rings In this section we review some useful properties of clean rings. We begin by showing that every homomorphic image of a clean ring is clean, that every direct product of clean rings is clean, and that every full matrix ring Mn(R) with entries from a clean ring R is clean. Then we consider the notion of lifting idempotents, and we show that a ring R is clean if and only if for any ideal I of R such that I \subseteq J(R) the quotient ring R=I is clean and idempotents lift modulo I. These properties are well known, but proofs are included for the sake of completeness. Basic Properties The next two results were proved by Anderson and Camillo [1]. 2 (1.1) Proposition. Every homomorphic image of a clean ring is clean. Proof. Since multiplication is preserved by every ring homomorphism, the homomorphic image of a unit (resp. idempotent) is a unit (resp. idempotent) of its ring. Since addition is also preserved by every ring homomorphism, the result follows. (1.2) Proposition. Every direct product of clean rings is clean. Proof. Since multiplication in a direct product of rings is defined componentwise, an element in a direct product of rings is a unit (resp. idempotent) of that ring if and only if the entry in each of its components is a unit (resp. idempotent) of its ring. Since addition in a direct product of rings is also defined componentwise, the result follows from a simple computation. A matrix with exactly one entry equal to 1 and all other entries equal to 0 is sometimes said to be a matrix unit and is denoted by Eij when the entry in the ith row and jth column is 1. Notice n that the set of n \times n matrix units fEiigi=1 is a finite set of mutually orthogonal idempotents inthe full matrix ring Mn(R) for any ring R, and notice that the sum E11 + E22 + \cdot \cdot + Enn is equal to the n \times n identity matrix. In general, a finite set of mutually orthogonal idempotents whose sumis equal to the identity 1 is said to be a complete set of orthogonal idempotents. The following result is due to Han and Nicholson [29]. (1.3) Theorem. Every full matrix ring Mn(R) with entries from a clean ring R is clean. Proof. Han and Nicholson [29, Theorem] showed that if the identity 1 of a ring R can be written as a finite sum 1= e1 + e2 + \cdot \cdot + en of mutually orthogonal idempotents ei such that each corner ring eiRei is clean, then the ring R is clean|the interested reader is encouraged to see their proof.