Ideals and Quotient Rings
Subject: ALGEBRA-III
Semester-IV
Lesson: Ideals and Quotient Rings
Lesson Developer: Divya Bhambri
College/Department: St. Stephen’s College,
University of Delhi
Institute of Life Long Learning, University of Delhi Page 1
, Ideals and Quotient Rings
Contents
1. Introduction ................................................................ 3
2. Ideals ......................................................................... 3
Examples of Ideals ............................................................ 3
Ideal Test ........................................................................ 5
3. Quotient Rings ............................................................. 8
4. Prime Ideals and Maximal Ideals .................................... 12
5. Principal Ideal Domain ................................................. 18
Exercises .......................................................................... 19
References........................................................................ 19
Suggested Readings ........................................................... 20
Institute of Life Long Learning, University of Delhi Page 2
, Ideals and Quotient Rings
1. Introduction
In this chapter, we define a quotient ring in a way analogous to the way in
which we defined quotient groups. The concept of an ideal is analogue of a normal
subgroup and helps us introduce the quotient rings. A quotient ring is also known
as Residue class ring or Factor ring. Further we establish the existence and find the
conditions under which quotient rings are integral domains or fields.
2. Ideals
We start this section by defining Ideals of a ring.
Definition: A subring 𝑆𝑆 of a ring 𝑅𝑅 is called a left ideal of 𝑅𝑅 if for every 𝑟𝑟 ∈ 𝑅𝑅 and
every 𝑎𝑎 ∈ 𝑆𝑆, we have 𝑟𝑟. 𝑎𝑎 ∈ 𝑆𝑆.
Definition: A subring 𝑆𝑆 of a ring 𝑅𝑅 is called a right ideal of 𝑅𝑅 if for every 𝑟𝑟 ∈ 𝑅𝑅 and
every 𝑎𝑎 ∈ 𝑆𝑆, we have 𝑎𝑎. 𝑟𝑟 ∈ 𝑆𝑆.
Definition: A subring 𝑆𝑆 of a ring 𝑅𝑅 is called a (two-sided) ideal of 𝑅𝑅 if for every
𝑟𝑟 ∈ 𝑅𝑅 and every 𝑎𝑎 ∈ 𝑆𝑆, we have both 𝑟𝑟. 𝑎𝑎, 𝑎𝑎. 𝑟𝑟 ∈ 𝑆𝑆.
Definition: An ideal 𝑆𝑆 of a ring 𝑅𝑅 is called a proper ideal of 𝑅𝑅 if 𝑆𝑆 is a proper
subset of 𝑅𝑅.
Value Addition
As clearly mentioned in the definition of an ideal, an ideal of a ring 𝑅𝑅 is always a
subring of 𝑅𝑅, whereas a subring neednot be an ideal of the ring.
Examples:
• Consider the ring (ℚ, + , . ).
Then it can be easily checked that (ℤ, +, . ) is a subring of (ℚ, + , . ).
But (ℤ, +, . ) is not an ideal of (ℚ, + , . ) as the product of a rational number
2 4
and an integer need not be an integer such as . 2 = ∉ ℤ.
3 3
• (ℝ , +, . ) is a subring of (ℂ , + , . ) that is not an ideal of (ℂ , + , . ). (Verify!)
Since the product of a real number and a complex number need not be a real
number.
Examples of Ideals:
1. Let 𝑅𝑅 be a ring. Then {0} & 𝑅𝑅 are ideals of 𝑅𝑅 known as the trivial ideals of 𝑅𝑅.
2. Consider the ring of integers ( ℤ, +, . ) and 𝐼𝐼 be the set of even integers, then
𝐼𝐼 is an ideal of ℤ.
Indeed, let 𝑥𝑥, 𝑦𝑦 ∈ 𝐼𝐼 and 𝑛𝑛 ∈ ℤ be arbitrary.
Then 𝑥𝑥 = 2𝑝𝑝 and 𝑦𝑦 = 2𝑞𝑞, for some 𝑝𝑝, 𝑞𝑞 ∈ ℤ.
Institute of Life Long Learning, University of Delhi Page 3
Subject: ALGEBRA-III
Semester-IV
Lesson: Ideals and Quotient Rings
Lesson Developer: Divya Bhambri
College/Department: St. Stephen’s College,
University of Delhi
Institute of Life Long Learning, University of Delhi Page 1
, Ideals and Quotient Rings
Contents
1. Introduction ................................................................ 3
2. Ideals ......................................................................... 3
Examples of Ideals ............................................................ 3
Ideal Test ........................................................................ 5
3. Quotient Rings ............................................................. 8
4. Prime Ideals and Maximal Ideals .................................... 12
5. Principal Ideal Domain ................................................. 18
Exercises .......................................................................... 19
References........................................................................ 19
Suggested Readings ........................................................... 20
Institute of Life Long Learning, University of Delhi Page 2
, Ideals and Quotient Rings
1. Introduction
In this chapter, we define a quotient ring in a way analogous to the way in
which we defined quotient groups. The concept of an ideal is analogue of a normal
subgroup and helps us introduce the quotient rings. A quotient ring is also known
as Residue class ring or Factor ring. Further we establish the existence and find the
conditions under which quotient rings are integral domains or fields.
2. Ideals
We start this section by defining Ideals of a ring.
Definition: A subring 𝑆𝑆 of a ring 𝑅𝑅 is called a left ideal of 𝑅𝑅 if for every 𝑟𝑟 ∈ 𝑅𝑅 and
every 𝑎𝑎 ∈ 𝑆𝑆, we have 𝑟𝑟. 𝑎𝑎 ∈ 𝑆𝑆.
Definition: A subring 𝑆𝑆 of a ring 𝑅𝑅 is called a right ideal of 𝑅𝑅 if for every 𝑟𝑟 ∈ 𝑅𝑅 and
every 𝑎𝑎 ∈ 𝑆𝑆, we have 𝑎𝑎. 𝑟𝑟 ∈ 𝑆𝑆.
Definition: A subring 𝑆𝑆 of a ring 𝑅𝑅 is called a (two-sided) ideal of 𝑅𝑅 if for every
𝑟𝑟 ∈ 𝑅𝑅 and every 𝑎𝑎 ∈ 𝑆𝑆, we have both 𝑟𝑟. 𝑎𝑎, 𝑎𝑎. 𝑟𝑟 ∈ 𝑆𝑆.
Definition: An ideal 𝑆𝑆 of a ring 𝑅𝑅 is called a proper ideal of 𝑅𝑅 if 𝑆𝑆 is a proper
subset of 𝑅𝑅.
Value Addition
As clearly mentioned in the definition of an ideal, an ideal of a ring 𝑅𝑅 is always a
subring of 𝑅𝑅, whereas a subring neednot be an ideal of the ring.
Examples:
• Consider the ring (ℚ, + , . ).
Then it can be easily checked that (ℤ, +, . ) is a subring of (ℚ, + , . ).
But (ℤ, +, . ) is not an ideal of (ℚ, + , . ) as the product of a rational number
2 4
and an integer need not be an integer such as . 2 = ∉ ℤ.
3 3
• (ℝ , +, . ) is a subring of (ℂ , + , . ) that is not an ideal of (ℂ , + , . ). (Verify!)
Since the product of a real number and a complex number need not be a real
number.
Examples of Ideals:
1. Let 𝑅𝑅 be a ring. Then {0} & 𝑅𝑅 are ideals of 𝑅𝑅 known as the trivial ideals of 𝑅𝑅.
2. Consider the ring of integers ( ℤ, +, . ) and 𝐼𝐼 be the set of even integers, then
𝐼𝐼 is an ideal of ℤ.
Indeed, let 𝑥𝑥, 𝑦𝑦 ∈ 𝐼𝐼 and 𝑛𝑛 ∈ ℤ be arbitrary.
Then 𝑥𝑥 = 2𝑝𝑝 and 𝑦𝑦 = 2𝑞𝑞, for some 𝑝𝑝, 𝑞𝑞 ∈ ℤ.
Institute of Life Long Learning, University of Delhi Page 3
