Let $A$ be a ring and $m_1,...,m_k$ maximal ideals of $A$, not necessarily different, and $F_i=m_1\cdots m_{i-1}/m_1\cdots m_i$. Because $m_iF_i=0$, $F_i$ can be made into a $A/m_i$-module defining the multiplication: $\hat{a}x=ax$, which is well-defined.
My question is:
Why is the lattice of $A$-submodules of $F_i$ the same as the lattice of $A/m_i$-submodules of $F_i$?
I'm trying to understand a proof for Hopkins-Levitzki Theorem. Sorry for my English.
In general, let $A$ be a ring, $I\subset A$ an ideal, and $M$ a $A/I$-module.
Then $M$ is obviously also a $A$-module, and a subgroup $N\subset M$ is a $A/I$-submodule iff it is a $A$-submodule.
Indeed, by definition $A\cdot N = (A/I)\cdot N$ because the action of $A$ is the same as the action of $A/I$, so $A\cdot N\subset N$ is equivalent to $(A/I)\cdot N\subset N$.
So the lattices of submodules are the same.