Abstract algebra

Question

Prove that if R is a ring and I ▹ R, then R[x]/I [x] ≃ (R/I )[x] for any indeterminate x.

Answer 1

Consider the map f : R[x] -> (R/I)[x] defined by the equation f(∑a_i xⁱ) = ∑ (a_i + I) xⁱ. Verify the f is a surjective ring homomorphism. The kernel of f is the set of all polynomials ∑ a_i xⁱ in R[x] such that ∑ (a_i + I) xⁱ = 0. Since ∑ (a_i + I) xⁱ = 0 if and only if a_i + I = I for all i, i.e., a_i ∈ I for all i, it follows that ker(f) = { ∑a_i xⁱ ∈ R[x] : a_i ∈ I for all i} = I[x]. By the first isomorphism theorem, we deduce R[x]/I[x] ≈ (R/I)[x].