# Submodules and Quotient Modules

A submodule $$M'$$ of a $$R$$-module $$M$$ is a subgroup of $$M$$ that is closed under scalar multiplication. The quotient group $$M/M'$$ becomes an $$R$$-module by defining $$a(x+M') = a x + M'$$. The $$R$$-module $$M/M'$$ is the quotient of $$M$$ by $$M'$$.

The natural map $$M \rightarrow M/M'$$ given by $$x \rightarrow x + M'$$ is a surjective module homomorphism, and it induces a bijection between submodules of $$M/M'$$ and submodules of $$M$$ that contain $$M'$$.

Let $$f:M\rightarrow N$$ be a module homomorphism. The kernel of $$f$$

$ker f = \{ x\in M | f(x) = 0 \}$

is a submodule of $$M$$. The image of $$f$$ is

$im f = f(M) = \{f(x) | x\in M\}$

is a submodule of $$N$$. The cokernel of $$f$$ is

$coker f = N / im f$

Let $$M'$$ be a submodule of $$M$$ contained in $$ker f$$. Then $$f$$ induces a homomorphism $$\bar{f} : M/M' \rightarrow N$$ given by $$x+M' \mapsto f(x)$$. Note $$ker \bar{f} = ker f / M'$$. If $$M' = ker f$$ we have the fundamental homomorphism theorem for modules:

$M/ker f \cong im f$

Ben Lynn blynn@cs.stanford.edu 💡