# 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.

Ben Lynn blynn@cs.stanford.edu 💡