Algebra

This book has been carefully designed in accordance with the model syllabus prescribed by the University Grants Commission (UGC), India, for courses in Algebra and Linear Algebra. It is well-suited for undergraduate and postgraduate students of mathematics across Indian universities and other institutions offering similar curricula. To ensure comprehensive understanding, the book begins with a preliminary chapter on Set Theory, providing the foundational concepts necessary for a deeper grasp of abstract algebraic structures. The content then progresses to cover a wide range of essential topics including integers, groups, rings, and fields, as well as polynomials, vector spaces, linear transformations, matrices, and Boolean algebra. Written in a clear and accessible language, the book emphasizes conceptual clarity through illustrative examples and real-world applications. Special attention has been given to demonstrating how abstract algebraic concepts are relevant and useful in number theory and the theory of equations, particularly in understanding the roots of polynomials. To enhance the utility of the text, an additional chapter on fuzzy set theory has been included. This modern topic introduces students to a broader perspective on set-related concepts and logical structures, offering them insight into emerging mathematical frameworks. Overall, the book serves as a valuable and self-contained resource for academic learning and exam preparation in algebra and linear algebra.