It is known that Artinian (commutative) ring has finitely many maximal ideals.
(One proof is by considering finite intersections of maximal ideals, and a chain from it, then using Artinian property of ring and some lemma.)
I have one problem in following arguments given in Mastumuta's Ring theory.
Let $\mathfrak{p}_1,\mathfrak{p}_2\cdots$ be infinite family maximal ideals of Artinian ring $R$.
Then the chain $$\mathfrak{p}_1 \supset \mathfrak{p}_1\mathfrak{p}_2 \supset \cdots$$ is strictly decreasing (why it is strictly decreasing?)
I tried to use Nakayama lemma, but we do not know whether ideals are finitely generated - since at this argument of Matsumuta, it is not known that ideals are finitely generated. (At the end of proof, it is proved that ideals are finitely generated.)
In short, I want to prove that above chain of ideals is strictly decreasing, without assuming them to be finitely generated. How can we prove it?
If $\mathfrak p_1\cdots \mathfrak p_k=\mathfrak p_1\cdots \mathfrak p_{k-1}$, then $\mathfrak p_1\cap\cdots\cap \mathfrak p_k=\sqrt{\mathfrak p_1\cdots \mathfrak p_k}=\sqrt{\mathfrak p_1\cdots \mathfrak p_{k-1}}=\mathfrak p_1\cap\cdots\cap \mathfrak p_{k-1}$. In other words, $\mathfrak p_k$ is a prime ideal containing $\bigcap_{i=1}^{k-1} \mathfrak p_i$. But this is the case if and only if $\mathfrak p_k\supseteq \mathfrak p_j$ for some $j\le k-1$.