The Isomorphism Theorems for Modules

#module_theory

As with groups and rings, for modules we have the standard four isomorphism theorems, the first of which gets by far the most use:

The First Isomorphism Theorem for Modules

Let $f : M \to N$ be an $R$ -module morphism. Then the map $m + \ker (f) \mapsto f (m)$ defines a module isomorphism $M / \ker (f) ≃ im (f)$ .

The proof of this theorem is almost identical to the proof of the analogous theorem for groups. Nevertheless, I should add a proof here at some point. In fact, I should provide an "element-free" proof that exclusively relies on invoking universal properties of the various constructions involved. For now, I'll mark that task as TBD.

The Second Isomorphism Theorem for Modules

Suppose $A, B$ are submodules of an $R$ -module $M$ . Then $(A + B) / B ≃ A / (A \cap B)$ .

See here for the definition of the sum of two submodules of a given module.

The Third Isomorphism Theorem for Modules

Suppose $A, B$ are submodules of an $R$ -module $M$ and $A \subseteq B$ . Then $(M / A) / (B / A) ≃ M / B$ .

Let's see how we can prove this theorem using only universal properties. To do that, we'll show that the module $M / B$ satisfies a universal property of the module $(M / A) / (B / A)$ , from which it will follow that the two are isomorphic (by a unique isomorphism!). With that in mind, let $π_{A} : M \to M / A$ and $π_{B} : M \to M / B$ denote the canonical projection morphisms (that are part of the information of the quotient). Since $\ker (π_{B}) = B$ and $A \subseteq B$ , by the universal property of $M / A$ the morphism $π_{B}$ factors uniquely through $π_{A}$ :

We now have our morphism $π_{A, B} : M / A \to M / B$ . At the level of elements, this is simply the map that takes each coset $m + A$ to the coset $m + B$ . Notice that $\ker (π_{A, B})$ is exactly the image of $B$ in $M / A$ , i.e., is $B / A$ . We're already in excellent shape.

Next, suppose $P$ is a module equipped with a morphism $f : M / A \to P$ such that $B / A$ is contained in $\ker (f)$ . Then the composition $π_{A} \circ f$ is a morphism from $M$ to $P$ such that $B \subseteq \ker (π_{A} \circ f)$ . By the universal property of $π_{B} : M \to M / B$ it follows that $π_{A} \circ f$ factors uniquely through $π_{B}$ (and hence $f$ factors uniquely through $π_{A, B}$ ):

We have therefore shown that every module morphism $f : M / A \to P$ with $B / A \subseteq \ker (f)$ factors uniquely through $π_{A, B}$ , which is precisely the universal property of the quotient $(M / A) / (B / A)$ (with its canonical projection morphism).

The Fourth Isomorphism Theorem for Modules

Let $N$ be a submodule of an $R$ -module $M$ . Then there is an isomorphism between the lattice of submodules of $M / N$ and the lattice of submodules of $M$ that contain $N$ .

Suggested next notes

Universal Properties I - Inspiring Examples
Direct products of modules
Direct sums of modules
Sums of submodules