Math 202 - Assignment 7 Solutions

Exercise 10.3.2. Let R be a commutative ring with identity. For all positive integers n and m, R ∼= R if and only if n = m. Proof. Let φ : R → R be an isomorphism of R-modules and let I E R be a maximal ideal. Then the map φ̄ : R → R/IR given by φ̄(α) = φ(α) is a morphism of R-modules. Moreover ker φ̄ = {α ∈ R | φ̄(α) = 0} = {α ∈ R | φ(α) ∈ IR} = φ−1(IRm) = IR. Therefore by the first isomorphism theorem R/IR ∼= R/IR. We already showed that R/IR ∼= (R/I). Since I is maximal, F := R/I is a field and we have an isomorphism of F -vector spaces R/I) ∼= (R/I). Hence n = m.