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 💡