Quotient modules

#module_theory

Since we've opened the door to studying subobjects, we can't shut that door before also allowing for the notion of quotient objects. (Don't blame me! I've already expressed my feelings on subobjects.)

Quotient modules

Existence of quotient modules

Suppose $R$ is a ring, $M$ is an $R$ -module and $N \subseteq M$ is a submodule. Then there is an $R$ -module $M / N$ together with a module morphism $π : M \to M / N$ , such that $\ker (π) \subseteq N$ "universally so"; i.e., such that every module morphism $g : M \to P$ with $N \subseteq \ker (g)$ factors uniquely through $π$ :

A note on

\ker (π)

One can prove (directly from the "universal property") that we actually have $\ker (π) = N$ .

This might look a bit intense, but once we cover universal properties in category theory we will see how it captures the true essence of the quotient module. For now, we simply note the following comforting facts.

The abelian group structure on $M / N$ is the usual group quotient of the abelian group $M$ by the (automatically normal) subgroup $N$ . Put another way, if $U : R - Mod \to A b$ is the forgetful functor, then $U (M / N) ≅ U (M) / U (N)$ . As such, its elements correspond to the distinct cosets of $N$ in $M$ . We can represent each element of $M / N$ in the form $m + N$ for some $m \in M$ , with the understanding that two cosets $m_{1} + N$ and $m_{2} + N$ define the same element in $M / N$ exactly when $m_{1} - m_{2} \in N$ . Moreover, the additive operation on $M / N$ is exactly coset addition: $(m_{1} + N) + (m_{2} + N) = (m_{1} + m_{2}) + N$ .
The action of $R$ on the $M / N$ is inherited from the action of $R$ on $M$ . In other words, for $r \in R$ and $m + N$ in $M / N$ we have $r (m + N) = r m + N$ . Convince yourself this is actually well defined!
The projection map $π : M \to M / N$ is the same as the projection map for the abelian groups, i.e., $π (m) = m + N$ . The key fact is that this is an $R$ -module morphism: $π (r m) = r m + N = r (m + N) = r π (m)$ .

From the construction of $M / N$ it is immediate that $\ker (π) = N$ , but one can prove that this equality is a direct consequence of the universal property. Try it!

By the way, the proof of this universal property of the quotient module is almost immediate from the corresponding universal property for quotient groups in the category $Ab$ .

Suggested next notes

The Isomorphism Theorems for Modules
Direct products of modules
Direct sums of modules
Sums of submodules