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Sachin Kaushal, Lovely Professional University Unit 15: Subrings

Unit 15: Subrings Notes

CONTENTS
Objectives

Introduction
15.1 Subrings
15.2 Summary
15.3 Keyword
15.4 Review Questions

15.5 Further Readings

Objectives

After studying this unit, you will be able to:

Discuss examples of subrings and ideals of some familiar rings

Explain whether a subset of a ring is a subring or not

Describe whether a subset of a ring is an ideal or not

Define and give examples of quotient rings

Introduction

In this unit, we will study various concepts in ring theory corresponding to some of those that
we have discussed in group theory. We will start with the notion of a subring, which corresponds
to that of a subgroup, as you may have guessed already.
Then we will take a close look at a special kind of subring, called an ideal. You will see that the
ideals in a ring play the role of normal subgroups in a group. That is, they help us to define a
notion in ring theory corresponding to that of a quotient group, namely, a quotient ring.
After defining quotient rings, we will look at several examples of such rings. But you will only
be able to realise the importance of quotient rings in the future units.

We hope that you will be able to meet the following objectives of this unit, because only then
you will be comfortable in the future units of this course.

15.1 Subrings

In last unit we introduced you to the concept of subgroups of a group. In this unit we will
introduce you to an analogous notion for rings. Remember that for us a ring means a
commutative ring.
In the previous unit you saw that, not only is Z  Q, but Z and Q are rings with respect to the
same operations. This shows that Z is n subring of Q, as you will now realise.

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