Let R be a ring φ : R → R be a ring homomorphism. Let S := {x ∈ R | φ(x) = x}.Prove that S is a subring of R.

Question

Let R be a ring φ : R → R be a ring homomorphism. Let S := {x ∈ R | φ(x) = x}.Prove that S is a subring of R.
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Solution 1

To prove that ( S ) is a subring of ( R ), we need to show that ( S ) is closed under addition, multiplication, and contains the additive identity and additive inverses.

  1. Additive Identity:
    • We need to show that ( 0 \in S ).
    • Since ( \phi ) is a ring homomorphism, ( \phi(0 Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study prob
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solv

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