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

\[ (M_1 + M_2)/M_1 \cong M_2 / M_1 \cap M_2 \]

Proof:

  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 follows by the fundamental homomoprhism theorem.

  2. 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

\[ (N:P) = \{a \in R | a P \subset N\} \]

(similar to ideal quotients). It is an ideal of $R$. Define the annihilator of $M$ by

\[ \mathrm{Ann}(M) = (0 : M) \]

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 well-defined 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:

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

  2. $ (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 finitely-generated.