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:

\[ M/ker f \cong im f \]