Basics of Modern Abstract Algebra

Abstract algebra is one of the fundamental branches of mathematics. It provides 
a systematic framework for studying algebraic structures and the relationships 
among them. The concepts introduced in this subject are essential not only for 
advanced mathematics but also for applications in computer science, coding 
theory, cryptography, physics and other scientific disciplines. 
This book, Basics of Modern Abstract Algebra, has been prepared primarily for 
undergraduate students studying mathematics under the UGC NEP 2020 curric
ulum. It is intended to provide a clear, logically organized and accessible intro
duction to the basic concepts of abstract algebra. The presentation begins with 
elementary ideas and gradually develops the subject through definitions, exam
ples, theorems, proofs and exercises. 
The book is organized into four chapters. Chapter 1 introduces sets, subsets, 
mappings, families of sets, equivalence relations and the properties of integers 
required for the subsequent study of algebra. Chapter 2 develops group theory, 
including subgroups, cosets, homomorphisms, permutation groups, symmetric 
groups, direct products, semidirect products and finite Abelian groups. Chapter 
3 discusses vector spaces, linear transformations, linear independence, bases and 
dual spaces. Chapter 4 presents rings and modules, including quotient rings, 
fields of quotients, principal ideal domains, polynomial rings and modules over 
principal ideal domains.  
The book may be used as a classroom textbook, a reference source or a guide 
for independent study. It is hoped that the systematic arrangement of the materi
al will help students develop mathematical maturity and prepare them for fur
ther study in algebra and related areas.