Commutative Algebra - Operations on Submodules

Operations on Submodules

Let $M$ be a module. Let $X$ be a subset of $M$ . Then define $⟨ X ⟩$ 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 $\sum M_{i} = ⟨ union M_{i} ⟩$ .

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) ≅ L / M

2. If $M_{1}, M_{2}$ are submodules of an $R$ -module $M$ then

(M_{1} + M_{2}) / M_{1} ≅ M_{2} / M_{1} \cap M_{2}

Proof:

The map $θ : L / N \to L / M$ 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 $θ : M_{2} \to (M_{1} + M_{2}) / M_{1}$ given by $x \to x + M_{1}$ .

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

Ann (M) = (0 : M)

If $I$ is an ideal of $R$ contained in $Ann (M)$ 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 $Ann (M) = 0$ . Note $M$ is faithful as an $R / Ann (M)$ -module.

The following can be easily verified:

1. Ann(M+N) = Ann(M) \cap Ann(N)

2. (N:P) = Ann((N+P)/N)

Let $x \in M$ . Then define $R x$ to be $⟨ x ⟩$ , 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 $M = ⟨ X ⟩$ (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.