I am trying to follow my old lecturers notes for commutative algebra and in particular, looking at the proof that any Artinian ring is Noetherian. There are a couple steps that I don't really understand and it would be of great help if someone can explain what is going on here:
Suppose $R$ is Artinian. Let $\mathscr{M}_1, \mathscr{M}_2, \cdots, \mathscr{M}_n$ be the maximal ideals in $R$. Since all prime ideals are maximal we have that $\bigcap_{i=1}^{n} \mathscr{M}_i$ is equal to the nilradical of $R$. Since in an Artinian ring, the nilradical is nilpotent, $\exists k > 0$ such that $(\bigcap_{i=1}^{n} \mathscr{M}_i)^k = 0$. Therefore, we have that $$ (\mathscr{M}_1\mathscr{M}_2\cdots\mathscr{M}_n)^k = 0. $$
Consider the following descending chain of ideals:
$$ R \supseteq \mathscr{M}_1 \supseteq \mathscr{M}_1^2 \supseteq \cdots \supseteq \mathscr{M}^k \supseteq \mathscr{M}_1^k\mathscr{M}_2 \supseteq \cdots \mathscr{M}_1^k\cdots \mathscr{M}_n^k = 0.$$
So far, so good. Here are the next steps, which is where I am confused:
(1) Each successive quotient has the form $I / \mathscr{M}_iI$ for maximal ideals $\mathscr{M}_i$.
(2) Therefore each $I / \mathscr{M}_iI$ is an $R /\mathscr{M}_i$ vector space in which descending chain of subspaces terminates.
I don't understand what these quotients are, how we are getting them or what they look like?
I also am confused about how they are $R /\mathscr{M}_i$ vector spaces and how do we know that every descending chain of subspaces in them terminates.
I would be very grateful if someone could explain these steps to me.
The description of the chain is a little elliptic. Here are some details:
A typical element in the chain has the form $$\mathfrak m_1^k \mathfrak \,m_2^k\,\dotsm \mathfrak m_r^k \,\mathfrak m_{r+1 }^i\qquad(0\le i<k,\;1\le r<n)$$ and the quotient of two successive elements is either \begin{align} &&&\underbrace{\mathfrak m_1^k \mathfrak \,m_2^k\,\dotsm \mathfrak m_r^k \,\mathfrak m_{r+1 }^i}_{=\,I}\bigm/\mathfrak m_1^k \mathfrak \,m_2^k\,\dotsm \mathfrak m_r^k \,\mathfrak m_{r+1}^{i+1}=I/I\mathfrak m_{r+1}\\ &\text{or }\qquad&& \underbrace{\mathfrak m_1^k \mathfrak \,m_2^k\,\dotsm \mathfrak m_r^k}_{=\,I}\bigm/\mathfrak m_1^k \mathfrak \,m_2^k\,\dotsm \mathfrak m_r^k \,\mathfrak m_{r+1}=I/I\mathfrak m_{r+1} \end{align} Both types are annihilated by $\mathfrak m_{r+1} $, so they're $R/\mathfrak m_{r+1}$ vector spaces. They satisfy the d.c.c. because $I$ satisfies the d.c.c. ( submodules of an artinian module) and a quotient of an artinian modules is an artinian module.