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If you want to show that every finitely generated $\mathbb Z[X]$-module is a direct sum of cyclic $\mathbb Z[X]$-modules, then forget it! The ideal $I=(2,X)$ can't be written as a direct sum of cyclic $\mathbb Z[X]$-modules (why?).
The ideal $I=(2,X)$ of $\mathbb Z[X]$ is not a direct sum of (non-zero) cyclic $\mathbb Z[X]$-modules. Let's suppose that $I=(2,X)$ is a direct sum of (non-zero) cyclic $\mathbb Z[X]$-modules. Then there exists a family $(N_{\alpha})_{\alpha\in A}$ of (non-zero) cyclic submodules of $I$ such that $I=\sum_{\alpha\in A}N_{\alpha}$, and ...