In Matsumura's commutative algebra, proposition 2.C, one says :
"A ring $A$ is artinian iff the length of $A$ as $A$-module is finite".
In the proof we have a descending chain : $ A \supseteq p_1 \supseteq p_1p_2 \supseteq .... \supseteq p_1... p_{r-1} \supseteq I \supseteq Ip_1 \supseteq Ip_1p_2 \supseteq ... \supseteq I^s =(0). $ where $p_1$ ...$p_r$ the maximal ideals of A and $I=p_1...p_r$. Can someone explain me the following sentence : "Each factor module of this chain is a vector space over the field $A/p_i=k_i$ for some $i$, and its subspaces correspond bijectvely to the intermediate ideals." ... Thank you.