Page 175 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 17: Ring Homomorphisms

CONTENTS
Objectives

Introduction
17.1 Homomorphisms
17.2 Properties of Homomorphisms
17.3 The Isomorphism Theorems
17.4 Summary

17.5 Keyword
17.6 Review Questions
17.7 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss whether a function is a ring homomorphism or not

Explain the kernel arid image of any homomorphism

Explain examples of ring homomorphisms and isomorphisms

Prove and use some properties of a ring homomorphism; state, prove and apply the

Fundamental Theorem of Homomorphisms for rings
Introduction

You have studied about the functions between groups that preserve the binary operation. You
also saw how useful they were for studying the structure of a group. In this unit, we will discuss
functions between rings which preserve the two binary operations. Such functions are called
ring homomorphisms. You will see how homomorphisms allow us to investigate the algebraic
nature of a ring.

If a homomorphism is a bijection, it is called an isomorphism. The role of isomorphisms in ring
theory, as in group theory, is to identify algebraically identical systems. That is why they are
important. We will discuss them also.
Finally, we will show you the interrelationship between ring homomorphism, ideals and
quotient rings.
17.1 Homomorphisms

Analogous to the notion of a group homomorphism, we have the concept of a ring
homomorphism. Recall that a group homomorphism preserves the group operation of its domain.
So it is natural to expect a ring homomorphism to preserve the ring structure of its domain.
Consider the following definition.

168 LOVELY PROFESSIONAL UNIVERSITY