Example of finitely generated Z[x]-module which is not a direct sum of cyclic modules 
Could you give an example of finitely generated Z[x]-module which is not a direct sum of cyclic modules? 

I have no idea about the example, could you give me some ideas? Thank you.
 A: Consider the ideal $I=(2,X)$ in $\Bbb Z[X]$.
Concretely, it consists of polynomials with integer coefficients of the form $a_0+a_1X+\cdots$ with $a_0$ even.
It cannot be generated by a single element $P(x)$ because you can never get $2$ and $X$ both multiples of a single polynomial.
But it is also false that $I=(2)\oplus(X)$ because $(2)\cap(X)$ contains non-zero elements (i.e. $2X$). A similar argument works if you attempt in any other way to generate $I$ with more than one generator.
The situation can be replicated almost verbatim for any ring which admits non principal ideals.
A: 
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 $N_{\beta}\cap\sum_{\alpha\ne\beta}N_{\alpha}=0$ for all $\beta\in A$. In particular, $N_{\alpha}$ are principal ideals in $\mathbb Z[X]$. But $N_{\alpha}\cap N_{\beta}\ne 0$ for $\alpha\ne\beta$ (if $x_{\alpha}\in N_{\alpha}$ and $x_{\beta}\in N_{\beta}$, then $x_{\alpha}x_{\beta}\in N_{\alpha}\cap N_{\beta}$). This leads us to the conclusion that $|A|=1$, that is, $I$ is principal, a contradiction.
