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MathidiaMathematics Academy
Rings and fields
Lesson 17 of 75
Easy practice

Ideals and quotients

Objective. Construct quotient rings from ideals.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

An ideal I of a ring R is an additive subgroup such that rI⊆I and Ir⊆I for all r in R, meaning I absorbs multiplication by any ring element, not just elements of I. Given an ideal I, the quotient ring R/I consists of cosets r+I with operations (r+I)+(s+I)=(r+s)+I and (r+I)(s+I)=rs+I, well-defined precisely because I absorbs multiplication. For example, in ℤ the ideal 5ℤ gives the quotient ring ℤ/5ℤ, which has exactly 5 elements {0+5ℤ,1+5ℤ,2+5ℤ,3+5ℤ,4+5ℤ} and is isomorphic to the field of integers mod 5.

Worked example

How many elements does the quotient ring ℤ/7ℤ have, formed from the ideal generated by 7?

7 … 7
Illustrated card showing 7 … 7
  1. 1The ideal 7ℤ consists of all multiples of 7
  2. 2Cosets of 7ℤ in ℤ are determined by remainder upon division by 7
  3. 3Possible remainders are 0,1,2,3,4,5,6
  4. 4This gives 7 distinct cosets, so ℤ/7ℤ has 7 elements

Answer: 7

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Common mistake. Students treat any subring as automatically an ideal, but ℤ is a subring of ℚ while not being an ideal of ℚ, since multiplying 1∈ℤ by 1/2∈ℚ gives 1/2∉ℤ, violating absorption.

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