Ring Theory & Linear Algebra I by Ranvendra Kumar is a systematically designed textbook tailored to the CBCS syllabus for B.Sc. 4th Semester Mathematics Honours students across Indian universities. Published by Mahaveer Publications, the book provides a strong conceptual foundation in two core areas of abstract algebra—Ring Theory and Linear Algebra—making it ideal for classroom teaching as well as self-study.

The book covers essential topics such as rings, subrings, integral domains, homomorphisms, ideals, factor rings, vector spaces, linear transformations, eigenvalues, eigenvectors, and matrix theory. Each chapter is presented in a clear, student-friendly manner with explanations, definitions, theorems, solved examples, and exercises that align with the CBCS curriculum.

UNIT 1 : RING THEORY

1. Ring

This section introduces the basic algebraic structure of rings and related concepts:

• Introduction to Rings

• Definition and Examples

• Integral Domain

• Field

• Skew Field / Division Ring

• Subring

• Subfield

• Characteristic of a Ring

• Characteristic of a Field

• Ordered Integral Domain

• Ordered Field

• Order Relation

• Ideals

• Simple Ring

• Smallest Left Ideal

These topics help students understand algebraic systems, their operations, and fundamental properties.

UNIT 2 : PRINCIPAL IDEAL & ISOMORPHISM

2. Principal Ideal and Isomorphism

A detailed study of ideal theory and structural properties of rings:

• Ideal generated by a subset

• Principal Ideal

• Principal Ideal Ring

• Quotient Ring / Ring of Residue Classes

• Isomorphism of Rings

• Homomorphism of Rings

• Kernel of Ring Homomorphism

• Prime Ideal

• Maximal Ideal

• Embedding of Rings

This unit strengthens the conceptual foundation needed for abstract algebra and advanced ring theory.

UNIT 3 : VECTOR SPACE

3. Vector Space

A complete foundation of vector spaces and their structural components:

• Definition of Vector Space

• Null Space

• General Properties

• Vector Subspace

• Linear Combination

• Linear Span

• Linear Dependence & Independence

• Direct Sum of Two Vector Spaces

• Direct Sum & Disjoint Subspace

• Complementary Subspace

• Basis of Vector Space

• Finite Dimensional Vector Space

• Invariance of Number of Basis Elements

• Dimension of Finitely Generated Vector Space

• Extension Theorem

• Dimension of Subspace

• Dimension of Linear Sum

• Quotient Space

This unit prepares the student for deeper linear algebra, geometry, and functional analysis.

UNIT 4 : LINEAR TRANSFORMATION

4. Linear Transformation

This section explains the behavior of linear maps, their structure, and algebraic properties:

• Definition and Introduction

• Linear Operator

• Vector Space Isomorphism

• Some Other Transformations

• Properties of Linear Transformation

• Range and Null Space

• Rank and Nullity Theorem

• Product of Linear Transformations

• Linear Algebra or Algebra of Linear Transformations

• Singular and Non-Singular Transformations

• Invertible Linear Transformations

This component trains students to understand mappings between vector spaces and their algebraic significance.

UNIT 5 : MATRIX REPRESENTATION OF LINEAR TRANSFORMATION

5. Matrix Representation

This unit connects linear transformations with matrices:

• Introduction

• Change of Coordinate Matrix

• Transition of the Matrix

Students learn how linear maps can be represented and analyzed using matrix theory—an essential skill for applied mathematics, physics, and computer science.