bilinear map

Let \(R\) be a ring and \(M,N,L\) \(R\)-modules A bilinear map ϕ: M × N → L$ is a map such that for each \(m \in M, n \in N\) the induced maps

\begin{align*} \varphi(m,\cdot): N \rightarrow& L \\ \varphi(\cdot,n): M \rightarrow& L \end{align*}

are module homomorphisms or equivalently

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