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
2. If \(M_1, M_2\) are submodules of an \(R\)module \(M\) then
Proof:

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 follows by the fundamental homomoprhism theorem.

Apply the same reasoning to the map \(\theta:M_2 \rightarrow (M_1 + M_2)/M_1\) given by \(x \rightarrow x + M_1\).
Let \(I\) be an ideal of \(R\). Then define the product \(I M\) to be the set of all finite sums \(\sum a_i x_i\) where \(a_i \in I, x_i \in M\). It is a submodule of \(M\).
Let \(N, P\) be submodules of \(M\). Define
(similar to ideal quotients). It is an ideal of \(R\). Define the annihilator of \(M\) by
If \(I\) is an ideal of \(R\) contained in \(\mathrm{Ann}(M)\) then \(M\) can be viewed as an \(R/I\)module by defining \((x+I) m = x m\) for all \(x \in R, m \in M\). This map is welldefined since \(I M = 0\).
An \(R\)module is faithful if \(\mathrm{Ann}(M) = 0\). Note \(M\) is faithful as an \(R/\mathrm{Ann}(M)\)module.
The following can be easily verified:

\( \mathrm{Ann}(M+N) = \mathrm{Ann}(M) \cap \mathrm{Ann}(N) \)

\( (N:P) = \mathrm{Ann}((N+P)/N) \)
Let \(x \in M\). Then define \(R x\) to be \(\langle x \rangle\), that is the set of all \(a x\) where \(a \in R\). A set \(X \subset M\) is a set of generators of \(M\) if \(M = \langle X \rangle\) (so every element of \(M\) can be written as a linear combination of elements of \(X\)). If \(M\) has a finite set of generators then \(M\) is said to be finitelygenerated.