Submodules

#module_theory

I have certain opinions on whether subobjects should really be studied (at least early on in a theory, and possibly ever). While category theory can certainly be forced to incorporate such a notion, it always seems (at least to me) exactly like that: forcing the theory to accommodate a concept for which it was not naturally built. I've also yet to encounter a situation in which the notion of subobject was strictly necessary and couldn't be replaced by a closely related concept (e.g., simply a monic arrow).

Nevertheless, for now I shall bow to the unstoppable collective compulsion to immediately introduce the notion of subobject after defining a new type of object. I will, however, frequently ask the question "Did we really need the notion of subobject here?" I think we'll find that the answer is usually "No, not really."

Definition of submodule

Let $R$ be a ring and $M$ be an $R$ -module. An $R$ -submodule of $M$ is a subset $N \subseteq M$ that is a subgroup (under the abelian group structure) and that is closed under the action of ring elements, i.e., $r n \in N$ for every $r \in R$ and $n \in N$ .

Note that, since abelian groups must be nonempty, so must submodules.

As usual in algebra, we have an equivalent criterion for checking to see if a subset of a module is a submodule.

The Submodule Criterion

Let $R$ be a ring and $M$ be an $R$ -module. A subset $N \subseteq M$ is a submodule of $M$ if and only if $N$ is nonempty and $n_{1} + r n_{2} \in N$ for every $r \in R$ and $n_{1}, n_{2} \in N$ .

Do I personally ever use this criterion? I do not. Please do not shame me.

Examples

The largest and smallest submodules

For each module $M$ , the entire module $M$ itself is a submodule, called the improper submodule. At the other extreme, the trivial subgroup ${0_{M}} \subseteq M$ is also a submodule, called the trivial submodule.

Submodules and ideals

When a ring $R$ is considered as a left $R$ -module (via left multiplication), the submodules of $R$ are in (natural) bijection with the left ideals of $R$ .

Submodules and subgroups

Recall that a $Z$ -module is "the same" as an abelian group. Under this identification, submodules are in (natural) bijection with subgroups.

Submodules and vector spaces

Recall that when $R = k$ is a field, modules are the same as $k$ -vector spaces. In this case, submodules are exactly the same as subspaces.

Submodules of $k [x]$ -modules

Recall that if $k$ is a field, then an $k [x]$ -module consists of a $k$ -vector space $V$ together with a $k$ -linear endomorphism $T : V \to V$ . Under this identification, $k [x]$ -submodules of $(V, T)$ are the $T$ -stable subspaces of $V$ , i.e., subspaces $U \subseteq V$ such that $T (U) \subseteq U$ .

Summary

$k [x]$ -submodules correspond to $T$ -stable subspaces, where $T$ is the $k$ -linear endomorphism encoding the action of $x$ .

Suggested next notes

Module morphisms and submodules
Quotient modules