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MathidiaMathematics Academy
Rings and fields
Lesson 16 of 75
Introduction

Rings and subrings

Objective. Verify ring structure and recognise examples.

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

Learn: the key idea

A ring is a set R with two operations, addition and multiplication, where (R,+) is an abelian group, multiplication is associative, and multiplication distributes over addition from both sides; a subring is a subset closed under these operations containing the additive identity. Rings may or may not have multiplicative identity 1 or be commutative, and checking the subring criteria (closure under subtraction and multiplication, containing 0) verifies substructures quickly. For example, 2ℤ={...,-4,-2,0,2,4,...} is a subring of ℤ since differences and products of even numbers remain even, even though 2ℤ lacks a multiplicative identity.

Worked example

Which of the following is a subring of ℤ: 3ℤ or the set of odd integers?

Options
  • 3ℤ
  • the set of odd integers
  • Both
  • Neither
  1. 1Check 3ℤ: sums, differences, and products of multiples of 3 remain multiples of 3
  2. 23ℤ contains 0 and is closed under subtraction and multiplication, so it is a subring
  3. 3Check odd integers: sum of two odd integers like 1+3=4 is even, not odd
  4. 4Odd integers fail closure under addition, so only 3ℤ is a subring

Answer: 3ℤ

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Practice

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Common mistake. Students assume every subring of a ring with identity 1 must also contain 1, but 2ℤ is a subring of ℤ without containing 1; always verify identity membership separately rather than assuming inheritance.

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