Structure theorem for finitely-generated modules over PID is well-known fact. But is there similar theorems for modules(maybe finitely-generated) over noetherian or artin or some other 'good' rings? I particularly want to know it in case $M$ is graded or have finite length.
Things are not so simple in the generalcase. Even for Dedekind rings, which are very close to PIDs — rings of integers of algebraic number fields, are not necessarily PIDs, but they are always Dedekind rings. Any ideal of such a ring is the product of a finite number of prime ideals, and the decomposition is unique up to the order of factors.
As for PIDs a finitely generated module is the direct sum of its torsion submodule and a finitely generated torsion-free submodule. However the torsion-free part is no more a free submodule, but only a direct summand of a free module — which is known as a (finitely generated) projective module. For more details, you can take a look here.
Yoshitomo Baba and Kiyoichi Oshiro in their book "Classical Artinian Rings and Related Topics"in Theorem 3.1.12. verify that over a Harada ring, every right R-module can be expressed as a direct sum of an in- jective module and a small module. It may be heplfull for YOU.