Operations on Submodules  Let $M$ be a module. Let $X$ be a
subset of $M$. Then define $\langle X \rangle$ to be the
submodule of $M$ generated by $X$, that is, the intersection of
all submodules of $M$ containing $X$. (Note that the
intersection of modules is itself a module.)  Let $\{ M_i | i
\in I\}$ be a family of submodules of $M$ (for some indexing set
$I$). Define their sum as for ideals: $\sum M_i$ consists of all
finite sums $\sum x_i$ where $x_i \in M_i$ and almost all $x_i$
are zero. Note $\sum M_i = \langle union M_i \rangle$.  Thus the
submodules of $M$ form a complete lattice with respect to
inclusion (the glb is interesction and lub is the sum).
Isomorphism Theorems  1. If $L \supset M \supset N$ are
$R$-modules then  \[ (L/N)/(L/M) \cong L/M \]  2. If $M_1, M_2$
are submodules of an $R$-module $M$ then  \[ (M_1 + M_2)/M_1
\cong M_2 / M_1 \cap M_2 \]     1. The map $\theta:L/N
\rightarrow L/M$ given by $x+N \mapsto x+M$ is a surjective
$R$-module homomoprhism with kernel $M/N$, thus the result