Unit group of rings and group rings.

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Unit Group of Rings and Group Rings

By

POOJA YADAV

Department of Mathematics

Submitted

in 加切 liment of the requirements of the degree of Doctor of Philosophy

to the

Indian Institute of Technology Delhi

January 2010

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To Paren ts

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CERTIFICATE

I am satisfled that the thesis entitled 伽iL Groi甲of Rings and C'iり，切 Rings 1 )re- sented by IVIs. Pooja. Yadav (2005八'1AZ8115) is wortiw ofcoisideration foi・the award of the degree of Doctor of philOSOJ)1〕き・and is a record of the bonaffide research work carried out i〕う・her u1icler m11v guidance and supei・vision and the results contained iii it h卿e not been subnlitt（うcl iii I)a1t or full to an》・other universit》・01、 Institute for award of aiiv degree/dipiくC)fl1.

R,K. ^{し一一一一喝ユ■}

Prof. R. K. Sharma ( Supervisor) January 2010

Depai・tment of 八1 athematics

indian Institute of Technology Delhi

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Acknowledgements

First and foremost. I

功

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my sincerest gratitude to my snpeγ

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isor Prof R. K.

Sharna. whose encouru.'

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ruent. guidance and s

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ort

戸り

m the injtial to the final le

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el enabled rue to develop an unde

務

tanding of the subject. He has s't

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po

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ted me througho

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t fl7?] thesis

て，

iith his patience and t

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ledge whi

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t allo

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ing me the room to work in my own way. I a

加効碗。坑。

level of my Doctorate d

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re( to li

窃

^encouragement and effort. Without hirn this thesis. would not have heen completed or wrjtien.

I would like to e

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ss mg deep sense of gratitude to Dr. Pramod Kanwar. Ohio

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e

窃

ity-Zanes

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ille

：加

r providing me

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nflinchjng encouraqement and s

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port in

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arz ons

初

ays. J am pri

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il

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ed to ha

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e the opport

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inity to com.e in contact 'with him for academic discussions

・

I am thankful to lIT Delhi authorities for pz

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me the necessaγ

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?] facilities for smooth completion of my wo

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I wo

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Id li

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e to e

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end my

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preciation to ?fly SRC

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tudent Research Committee) members

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1 Wagish Shi.'kla and Prof I

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^thili Sharan (Cent eγ

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heric Sciences. IlT Delhi),as

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ell as to ail t/ze faculty

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ein be

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of the Department

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Mathematics. lIT Delhi for theiγ

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encouragement. I

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ould also like to thank C

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R for p

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riding . financial assistance by awarding me a research fellowship

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my Ph.D'

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earch work.

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de

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est

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γatitzide goes to my family

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their

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e and suppuri.

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11 ACKNOWLEDGEMENTS

th7りuqho'ut m.y i施：without them ihis dissertation is si，叩ly impossible. I am indebted to my fat/ic'r・foγ・his care and loりe. No切。価can be ever enough to thank my husband, Praveen. for his constant encouragement,unde摺tanding and endless patience. Ii勾 Ph.D. had always been ffi',説 p司erence for him. I o切e special thanks to my paルn加 my in-la初s and ali my otlieγかrn仰 'rnembcおfor their patience. io肥 and s堺)pOrt during these yea摺げruy studies.

I would like to thank all myかends.．加γtheir constant encouragement and s叩－

po元‘ I切ovid particula冗y like to thank Vandana and及,mit for being there to share theりarioて‘s highs and lows and numerous eて"Ps げcoi:元e at the e功配 sh叩，ぬapna and Sangccta for theかwond。巾i cori切αny e.specねuy面九ng my stay in 疏madri h昭－

tel. I don't /iaりe woγ需 to thank ray colleagues Bale/rand, Dhirendra. Dinu, Pvncet and ぬnil for providing mcαcoγゆ元αble and jo効il environment turo四ho厩 my Ph.D. I 叩preciate all my seni併、for rnα厩ng mc comfo冗abた in t/rC d叩αrtment and, .m,tivα加ig in the 加漉可 days げmy research. I 初illαんays remember 駈肥加 Bli.a四a，幼rut',. Anand. Alo/c for provi庇ng fun-filed and cheerful environment in the department.

I will never forget Kane/ran for lier genuine concern. She always boosted and s町切o元ed me selflessly. I also thank lier for countless i叩ror叩t'i,, academic and non-acadcmZc which we had：加rn v.'hich J have learnt a lot about 7でsearch. みzst saying than厨切ill be too little a gratitude towards her.

月naily, I thank the almighty Cod for his blessings.

New Delhi January 2010

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Abstract

The aim of the thesis is to study the unit group of rings and group rings. We focus mainly on the generators and pi・esentations of linear groups. An element a of a i・ing R is said t'o be unit iでguiar if a=aてi,a for sorne unit てL ifl R. Equivalently. a is unit regular if and only if a=eu foi、 sorne idempotent e and1 some unit u in R. An element a E R is called clean if a=e+1. for sorne idempotent e and some unit u in R. Here.we introduce the concept of Lie regular elements. An element a of a ring R is said to be a Lie regular if a=[ e, u]=eu一ue. where e is an idempotent in R and ?J. is a unit of R. Further. a unit in R is said to be Lie regular unit if it is Lie regn lai・as an element of R. In this thesis, we study Lie regulal・units and show the existence of Lie regular elements and Lie regular units in 2 x 2 matrix rings. Also.

we have shown that any Lie regular element in J石（R):where R is a comim〕utative domain in which 2 is invertible and in time gi・oup algebra KD叫 of inffinite dihedral grout〕over a fleld of characteristic not equal to 2, can be expressed as [v1,712],wh」ere 'a] and v,2 aiで mmits. V'厄 obtain Lie regulai・units as generators of linear groups GL(2. Z1>), CL(2. Z2・・）:GL(2，ろv),GL(2. Z2p・・）,GL(2,Z3・・） and GL(2. Z51，・・）,wh、ei・e p is any odd 1)rilne. Pi・esentations of linear gi・oups GL(2. Z4),GL(2:Z6):GL(2. Z8), CL(2,Z9), CL(2,Z10),GL(2, Z12),GL(2, Z14):GL(2, Z15), GL(2, Z22):GL(2. Z25), CL(2, Z26):GL(2. Z27) and GL(2. Z34) are also given in ternms of Lie regular units as generators.

111

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Iv ABSTRACT

v兆h卿e studied cii・culant matrices, their properties and equiぬlent results to obtain the structure of the unit group of integral group rings. We have given a complete characterization of the unit groups tl(C？・3（ろk)),U(C？・4cF;）り） and U(C？・。(Zr)）・

Further, we have obtained presentations of the unit groups tI(Z2Q8) and U(Z2D8).

We have described a complete characterization of the unit group 7(ZPQ8). We have given a matrix representation foi・the group algebra R(C2 x Doo). In pai・ticulai , we have shown that U(Z2(C2 x D叫)) is not ffinitely generated.

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"

v

i

n

nJ 1弓 QO 2 2 3 【ー一ーつ」 5 5 6 1

ユつ L 7 「ー Preliminaries

Unit Group of C，・^?'（凡）

『ー上つ」 4 Al

Index Bio-data

91 94

つ」りJ 7 8 8 8