Pathological examples of finitely generated modules

Question

Let $R$ be a commutative (noetherian) ring with identity. Let $M$ be a finitely generated $R$-module. Let $n\in\mathbb{N}$. Does $$M=\bigoplus_{k=1}^n M_k,$$ where $M_k$ is an either an ideal or quotient ring of $R$. If not, are there any explicit counterexamples and is there a better characterization of finitely generated modules over $R$?