Subring Test

Abstract Algebra · Axiom Academy

A simple two-step method to verify whether a subset of a ring forms a subring Closed under subtraction: For all a, b ∈ S, we have a - b ∈ S Closed under multiplication: For all a, b ∈ S, we have ab ∈ S 2. Example: Gaussian Integers ℤ[i] The Gaussian integers ℤ[i] = a + bi : a, b ∈ ℤ form a subring of the complex numbers ℂ. Let's verify using the Subring Test. 3. Example: Continuous Functions Let C[a,b] denote the set of all continuous functions from [a,b] to ℝ. This forms a subring of the ring of all functions F[a,b]. Consider the set of 2×2 diagonal matrices with integer entries. This forms a subring of M₂(ℤ), the ring of all 2×2 integer matrices. 5. Counter-Example: When Multiplication Fails Not every subset closed under subtraction is a subring! Consider the set S = 0, 2, 4, 6, ... of even integers in ℤ.

This is the written version of the interactive lesson above. See the full Abstract Algebra course.