Let ( R, J ) be a left noetherian local ring, and M be a...

Let ( $R, J$ ) be a left noetherian local ring, and $M$ be a finitely generated left $R$-module. Show that $M$ is a free $R$-module iff, for any exact sequence $0 \rightarrow A \rightarrow B \rightarrow M \rightarrow 0$ of left $R$-modules, the induced sequence $0 \rightarrow A / J A \rightarrow B / J B \rightarrow M / J M \rightarrow 0$ remains exact.

Let M and N be free left R-modules of finite rank. Suppose that N = N1 ⊕ N2: this means that N1 and N2 are two submodules of N such that N1 ∩ N2 = {0} and every element n of N can be expressed as n = n1 + n2 with n1 ∈ N1, n2 ∈ N2. (a) (2 points) Prove that N/N1 is isomorphic to N2. Let ψ : M → N be a homomorphism of R-modules. Set M1 = ψ^-1(N1) = {m ∈ M | ψ(m) ∈ N1}. M1 is a submodule of M. (You don't have to explain this.) (b) (3 points) Suppose that R is a principal ideal domain. Prove that M/M1 is a free R-module. Hint: If you denote by φ : N → N2 the homomorphism that you needed to use in part (a), then consider the homomorphism φ ∘ ψ : M → N2 from M to N2.

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