ABSTRACT ALGEBRA - University of Sulaimani

University of Sulaimani 

College of Science 

Department of Mathematics 

ABSTRACT ALGEBRA 

Course Coordinator 

Dr. Kawa Ahmed Hasan 

2010-2011

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Course Coordinator and the list of teachers on 

this course@ 

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@@Abstract Algebra @@@@@@@@Z‘ŠŽíØ@õìbä 

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@@Assistant Professor Dr. Adil Kadir Jabbar 

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@@Department of Mathematics-College of Science 

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@@HDr. Kawa Ahmed HasanI 

E-mail@Z@ðäŽìÙÝï÷@ñ‡îŠói 

@@adilkj@gmail.com 

kawa79b@gmail.com 

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Course Overview 

Abstract Algebra is one of the important subjects in mathematics it 

consists of three parts Group Theory, Ring Theory and Field Theory. 

The present course deals only with the Group Theory and Ring Theory. 

@@Part I : Group Theory 

The concept of the group has been introduced for the first time in 

1770 and extends to the twentieth century, but the major developments 

occurred in nineteenth century. This part will outline the origins of the 

main concepts, results, and theories discussed in a first course on group 

theory, for example, the concepts of abstract groups, normal subgroups, 

quotient groups, simple groups, free groups, isomorphisms, 

homomorphisms, automorphisms, composition series, direct products, 

the theorems of Lagrange, Cauchy, Cayley, Jordan-Holder, the theories 

of permutation groups and of abelian (commutative) groups. There are 

four major sources in the evolution of group theory, they are: 

1. Classical Algebra (Lagrange, 1770): 

Lagrange wrote his fundamental memoir " Reflections on the 

solution of algebraic equations" concerned polynomial 

equations. There were 

(i) Theoretical questions: dealing with the existence and nature of 

the roots, for example, does every equation have a root ? how 

many roots are there ? are they real, complex, positive, negative ? 

(ii) Practical questions: dealing with methods for finding the roots. 

In the later, there were exact methods and approximate methods. 

2. Number Theory (Gauss, 1801): 

Gauss summarized and unified much of the number theory that 

preceded him. The work also suggested new directions which kipped 

mathematician occupied for the entire century. The groups appeared 

in four different guises : the additive group of integers modulo n , 

that denoted by Z n , the multiplicative group of integers relatively 

prime to n , modulo n , the group of equivalence classes of binary 

quadratic forms, and the group of n − th roots of unity. 

3. Geometry (Klein, 1874): 

Klein has classified geometry as the study of invariants under 

various groups of transformations. Here there appear groups such

as the projective group, the group of rigid motions, the group of 

similarities, the hyperbolic group, the elliptic group as well as the 

geometries associated with them. 

4. Analysis (Lie, 1874 ; Poincare and Klein, 1876): 

In 1874, Lie introduced his general theory of continuous 

transformation groups- essentially what we call Lie groups today, 

while Poincare and Klein began their work on " Automorphic 

functions " and the groups associated with them around 1876. 

Automorphic functions (which are generalizations of he circular, 

hyperbolic, elliptic, and other functions of elementary analysis) 

are functions of a complex variable z and there is a very famous 

group which is referred to Klein as a Klein 4- group. 

@@Part II : Ring Theory 

Among the most important examples of rings are the integers, 

polynomials, and matrices. Simple extensions of these examples are at 

the roots of ring theory and thus we have the following three examples: 

1. The integers Z can be thought of as the appropriate subdomain of 

the field Q of rational numbers in which to do number theory. 

2. The polynomial rings R [x] 

and R [ x, 

y] 

in one and two variables, 

where R denotes the real numbers. In particular, while the roots of a 

polynomial in one variable constitute of discrete set of real numbers, 

the roots of a polynomial in two variables constitute of a curve in the 

plane a so-called algebraic curve. 

3. Square n × n matrices over the real numbers, can be viewed as n 2 - 

tuples of real numbers with coordinate wise addition and 

appropriate multiplication obeying the axioms of a ring. Our third 

n 

example consists of n − tuples R of real numbers with coordinate 

wise addition and appropriate multiplication, so that the resulting 

system is a (not necessarily commutative) ring. Such systems, often 

extensions of the complex numbers.

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Course Objectives 

The main aims of the present course are: 

1. To introduce the basic concepts of group theory and ring theory to 

the students and illustrate these concepts by examples. 

2. To give the ideas on the nature of some problems in group theory 

and ring theory and then giving the methods by which one can prove 

those problems. 

3. The concept of a group can be used to explain some geometrical and 

physical phenomena, so that there are so many applications of group 

theory in physics and chemistry and for this reason this subject is 

also necessary for physics and chemistry students. 

4. Ring theory has been constructed as a trial to abstract some known 

properties of integers and algebraic numbers, so that these numbers 

were (till to now) an important and a suitable area to the ideas on 

which the developments of ring theory are based. 

5. The first part of this course consists of five chapters. In the first 

chapter of this course the groups and subgroups are defined and so 

many examples of them are given to make students so much familiar 

with the nature of groups. The second chapter deals with some 

special types of groups and subgroups such as, cyclic groups, simple 

groups, symmetric group S n , abelian groups, normal subgroups, the 

center of the groups, the alternating group A n and the quotient 

groups. The third chapter is devoted to study the properties of 

groups and the ability of transferring these properties from a group 

to an other by a mapping which is known as a group 

homomorphism and at the end of this chapter some classical 

isomorphism theorems are given by which we can determine those 

groups which are isomorphic, so that they have exactly the same 

algebraic properties. In chapter four, we have presented prime 

groups and sylow p − groups and the sylow theorems are given 

which help us to know whether a finite group is simple or not and 

several examples of simple groups at the end of this chapter are 

given. The last chapter of this part deals with indecomposable and 

solvable groups. In fact, the concept of solvable groups gave the 

whole answer to a very old question which is: can we solve every 

equation

of degree n , for n ≥ 5, by radicals ?. 

6. The second part of this subject consists of eight chapters. In chapter 

six and chapter seven we gave the definitions of rings, subrings and 

ideals and supported with examples. Also, some types of rings and 

subrings are introduced such as, the center of a ring, maximal ideals, 

prime ideals, primary and semiprimary ideals and the relationships 

between these types of ideals are determined and the quotient rings, 

ring homomorphisms are studied in chapter eight and at the end of 

this chapter the concept of embedding of rings are introduced. 

Euclidean domains, principal ideal domains, unique factorization 

domains, prime elements and irreducible elements all are studied in 

chapter nine and chapter ten and some relations between these 

integral domains are obtained. In chapter eleven, we defined and 

studied polynomial rings and the relations between this type of rings 

with Euclidean domains, principal ideal domains and unique 

factorization domains are determined. Also, irreducible and 

primitive polynomials are studied. In chapter twelve, Noetherian 

and Artinian rings are studied and the relations between them are 

given. The last chapter was devoted for defining the nil radical of 

ideals and some results are proved.

A List of Some References: 

[1] S. Singh & Qazi Zameeruddin : ” Modern Algebra ” Vikas 

Publishing House ( 1972). 

[2] V. K. Khanna & S.K.Bhambri : ” A Course in Abstract 

Algebra ” Vikas Publishing House (2004). 

[3] H. A. Nielsen : ” Elementary Commutative Algebra ” 

Department of Mathematical Sciences-University of Aarhus 

(2005). 

[4] P. Garrett : ”Abstract Algebra-Lectures and Worked Examples 

for a Gradate Course ” (2005). 

[5] A. Hermann : ” Abstract Algebra ” (2004).

ABSTRACT ALGEBRA - University of Sulaimani

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