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submodule

Definition

Thoughout let $R$ be some ring. Write $R$Mod for the category of module over $R$. Write $UR\mathrm{Mod}\to$ Set for the forgetful functor that sends a module to its underlying set.

Definition

For $N\in R\mathrm{Mod}$ a module, a submodule of $N$ is a subset of $U(N)$ which

1. is a subgroup of the underlying group (closed under the addition in $N$);

2. is preserved by the $R$-action.

Equivalently this means:

Definition

A submodule of $N\in R\mathrm{Mod}$ is a module homomorphism $i:K\to N$ whose underlying map $U(i)$ of sets is an injection.

And since the injections in $R$Mod are precisely the monomorphisms, this means that equivalently

Definition

A submodule of $N\in R\mathrm{Mod}$ is a monomorphism $i:K\hookrightarrow N$ in $R$Mod. Hence a submodule is a subobject in $R$Mod.

Remark

Given a submodule $K\hookrightarrow N$, the quotient module $\frac{N}{K}$ is the quotient group of the underlying abelian groups.

Examples

Example

For $N=R$ regarded as a module over itself, a submodule is precisely an ideal of $R$.

Example

For $f:N_1\to N_2$ a homomorphism of modules,

1. the kernel $\ker (f)\hookrightarrow N_1$ is a submodule of $N_1$,

2. the image $\mathrm{im}(f)\hookrightarrow N_2$ is a submodule of $N_2$.

Remark

In example 2 quotient module of $N_2$ by the image is the cokernel of $f$

$$\mathrm{coker}(f)\simeq \frac{N_2}{\mathrm{im}(f)}.$$coker(f) \simeq \frac{N_2}{im(f)} \,.

Properties

Of free modules

Let $R$ be a ring.

Proposition

Assuming the axiom of choice, the following are equivalent

1. every submodule of a free module over $R$ is itself free;

2. every ideal in $R$ is a free $R$-module;

3. $R$ is a principal ideal domain.

A proof is in (Rotman, pages 650-651).

References

For instance

• Rotman Advanced Modern Algebra