Noether-Lasker Theorem

Let $M$ be a finitely generated Module over a commutative Noetherian Ring $R$. Then there exists a finite set $\{N_i\vert 1\leq i\leq l\}$ of submodules of $M$ such that

1. $\cap_{i=1}^l N_i=0$ and $\cap_{i\not=i_0} N_i$ is not contained in $N_{i_0}$ for all $1\leq i_0\leq l$.

2. Each quotient $M/N_i$ is primary for some prime $P_i$.

3. The $P_i$ are all distinct for $1\leq i\leq l$.

4. Uniqueness of the primary component $N_i$ is equivalent to the statement that $P_i$ does not contain $P_j$ for any $j\not=i$.