Question

For a finite abelian group $G$ is there an analogue of structure theorem for finitely generated modules like for P.I.D. rings but with $Z[G]$ group ring over integers instead ?

Answer 1

Let me give an answer in the opposite direction of the ones already given. It may that be you can't get a precise classification, but nevertheless, you can say something, and it may be enough for the situation you are considering, or at least helpful.

Let's suppose first that your module $M$ is torsion; then it is a product of its $p$-power torsion parts for each prime $p$, so you an suppose it is $p$-power torsion. It is then a module over $\mathbb Z_p[G]$. You may write $G$ as a product $G_1\times G_2,$ where $G_1$ is of prime-to-$p$ order and $G_2$ is of $p$-power order. The group ring $\mathbb F_p[G_1]$ is a product of finite fields $k$, and the ring $\mathbb Z_p[G_1]$ is isomorphic to a corresponding product of rings of Witt vectors $W(k)$. So one is left with $p$-power torsion modules over $W(k)[G_2]$. This may be something of a mess, but one can say something: for example, any Jordan--H\"older factor of such a module is the trivial representation of $G_2$ over $k$, and so your module is a successive extension of copies of the trivial representation.

Now suppose that you have an $M$ which is not necessarily torsion. Then one can consider the map $M \to \mathbb Q\otimes_{\mathbb Z} M$. The kernel is torsion, and so one can get some handle on its structure as above. The image is a $G$-invariant lattice in the $G$-representation $\mathbb Q \otimes_{\mathbb Z} M.$ The possibilities for this latter representation (of $\mathbb Q$ over $G$) can be described pretty easily, since $\mathbb Q[G]$ is just a product of fields $K$; it will break up into a product of copies of these various fields $K$. The biggest complication with the image of $M$ is that it typically won't break up as a product.

A useful way to think is to consider Spec $\mathbb Z[G]$, which is an affine scheme mapping finitely to Spec $\mathbb Z$, and then to think of $M$ as quasi-coherent sheaf over Spec $\mathbb Z[G]$. Then tensoring with $\mathbb Q$ corresponds to looking at this sheaf at the generic points; looking at the torsion part of $M$ corresponds to looking at sections supported at closed points; and the possible failure of $M$ to decompose as a product corresponds to the fact that although Spec $\mathbb Z[G]$ will have several irreducible components (if $G$ is non-trivial) --- these correspond to the various factors in the decomposition of $\mathbb Q[G]$ into a product of fields --- it will be connected (the components meet at various closed points, corresponding to the primes dividing the order of $G$).

Number theorists frequently work with rings like $\mathbb Z[G]$, and modules over them. Depending on what you want to do, it is not necessary to have a complete classification; a somewhat coarser understanding of the type described above can be quite useful by itself.

Answer 2

The answer to your question is 'no'. Even if you limit yourself to modules that are free over $\mathbb{Z}$, there is no classification known. Indeed, if your abelian group is not cyclic or its order is not cube-free, then there are infinitely many isomorphism classes of indecomposable $\mathbb{Z}$-free modules (this is due to many people). So it is certainly not true, that they are all quotients of the regular module, like in the PID-case.

Of course, it is still true that any finitely generated $\mathbb{Z}[G]$-module is a quotient of a free module, but that is general algebraic non-sense and doesn't tell you anything about their structure.

For a (admittedly slightly aged) survey of known results on $\mathbb{Z}$-free $\mathbb{Z}[G]$-modules, aka integral $G$-representations, see this survey.