Is every finitely-generated module over $\mathbb{Z}/4\mathbb{Z}$ is of the form: $(\mathbb{Z}/4\mathbb{Z})^a \times (\mathbb{Z}/2\mathbb{Z})^b$ A module over $\mathbb{Z}/4\mathbb{Z}$ is just an abelian group satisfying the identity $x+x+x+x=0$. Since $4$ isn't prime, hence $\mathbb{Z}/4\mathbb{Z}$ isn't a field (or even an integral domain): indeed, $2(\mathbb{Z}/4\mathbb{Z})$ is a non-trivial proper ideal, and the resulting quotient is isomorphic to $\mathbb{Z}/2\mathbb{Z}.$ This means that both $\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$ can be viewed as modules over $\mathbb{Z}/4\mathbb{Z}.$ It therefore seems possible that every finitely-generated module over $\mathbb{Z}/4\mathbb{Z}$ is of the form
$$(\mathbb{Z}/4\mathbb{Z})^a \times (\mathbb{Z}/2\mathbb{Z})^b$$
for appropriately chosen naturals $a$ and $b$.

Question. Is this true?
If so, how far can this be generalized?
If not, a counterexample is sought.

 A: A finitely generated module over $\Bbb Z/4\Bbb Z$ is also a finitely generated abelian group. All finitely generated abelian groups are isomorphic to a group of the form
$$
(\Bbb Z/p_1^{i_1}\Bbb Z)^{a_1}\times\cdots \times (\Bbb Z/p_n^{i_n}\Bbb Z)^{a_n} \times \Bbb Z^{a_{n+1}}
$$
where $p_j$ are distinct primes, and $a_j, i_j$ are natural numbers. Since we also need $x+x+x+x = 0$ to hold, none of the $p_j^{i_j}$ can be anything other than $2$ or $4$, and we must have $a_{n+1} = 0$. So yes, you have found all the finitely generated modules.
A: 
How far can this be generalized?

There is a quite old classical result (perhaps not too-well-known) that the rings whose modules are direct sums of cyclic submodules are precisely the Artinian principal ideal rings. $\Bbb Z/4\Bbb Z$, being finite and uniserial, is a simple example of this type of ring.
If I remember correctly, Carl Faith's books refer to a ring whose modules decompose into direct sums of cyclic submodules as $\Sigma$-cyclic. The Same condition relaxed to only finitely generated modules is called $\sigma-$cyclic.  I don't remember if there is a characterization of $\sigma$-cyclic rings, but I do believe Noetherian serial rings are among them.
