module homomorphism to the hom object induced bilinear map for a commutative ring

1. Proposition

Let \(R\) be a commutative ring, \(M,N,L\) \(R\)-modules and \(\mathrm{Hom}_{\mathrm{RMod}}(N,L)\) the inner hom. Then there exists a natural Bijection between

module-homomorphisms \(f: M \rightarrow \mathrm{Hom}_{\mathrm{RMod}}(N,L)\)
bilinear maps

\begin{align*} g: M \times N \rightarrow& L \\ \end{align*}

given by

\begin{align*} \varphi: \mathrm{Hom}_{\mathrm{RMod}}(M, \mathrm{Hom}_{\mathrm{RMod}}(N,L)) \rightarrow& \mathrm{Hom}_{\mathrm{bilinear}}(M \times N, L) \\ f \mapsto& \left( (m,n) \mapsto (f(m))(n) \right) \\ \varphi^{-1}: \mathrm{Hom}_{\mathrm{bilinear}}(M \times N, L) \rightarrow& \mathrm{Hom}_{\mathrm{RMod}}(M, \mathrm{Hom}_{\mathrm{RMod}}(N,L)) \\ ((m,n) \mapsto g(m,n)) \mapsto& (m \mapsto g(m,-_N) \end{align*}

2. Proof

2.1. "elementary bilinear" to morphism into hom

2.1.1. welldefined codomain

Note that for \(m \in M\), by definition

\begin{align*} g(m)(-) \in \mathrm{Hom}_{\mathrm{RMod}}(N,L) \end{align*}

is a module homomorphism

2.1.2. module homomorphism

Let \(m_1,m_2 \in M\) and \(r \in R\) and \(n\) a fixed \(n\) Then

\begin{align*} g((m_1 + r \cdot m_2),n) =& (f(m_1 + r \cdot m_2))(n) \\ =& (f(m_1) + r \cdot f(m_2))(n) \\ =& f(m_1)(n) + r \cdot f(m_2)(n) \\ =& g(m_1,n) + r \cdot g(m_2,n) \end{align*}

This shows that for each \(n \in N\)

\begin{align*} g(m_1 + r m_2,n) =& g(m_1)(n) + r \cdot g(m_2,n) \end{align*}

or equivalently

\begin{align*} g(m_1)(-) + r \cdot g(m_2)(-) \end{align*}

2.2. morphism into hom to "elementary bilinear"

Let \(m_1,m_2 \in M,n_1,n_2 \in N\) and \(r_1,r_2 \in R\). Then for a module homomorphism

\begin{align*} f: M \rightarrow \mathrm{Hom}_{\mathrm{RMod}}(N,L) \end{align*}

we get

\begin{align*} f(m_1 + r_1 \cdot m_2)(n_1 + r_2 n_2) =& f(m_1 + r_1 \cdot m_2)(n_1) + r \cdot f(m_1 + r_1 \cdot m_2)(n_2) \end{align*}

as \(f(m_1 + r_1 \cdot m_2)\) is a module homomorphism \(N \rightarrow L\).

Furthermore it holds by definition of a module hommorophism \(M \rightarrow \mathrm{Hom}_{\mathrm{RMod}}(N,L)\)

\begin{align*} f(m_1 + r_1 \cdot n_1)(-) =& f(m_1)(-) + r_1 \cdot f(m_2)(-) \end{align*}

showing

\begin{align*} f(m_1 + r_1 \cdot m_2)(n_1 + r_2 n_2) =& f(m_1 + r_1 \cdot m_2)(n_1) + r \cdot f(m_1 + r_1 \cdot m_2)(n_2) \\ =& f(m_1)(n_1) + r_1 \cdot f(m_2)(n_1) + r_2 \cdot f(m_1)(n_2) + r_1 r_2 \cdot f(m_2)(n_2) \end{align*}