---- > [!definition] Definition. ([[bilinear map]]) > Let $R$ be a [[commutative ring|commutative]] [[ring]] and $M,N,P$ be $R$-[[module|modules]]. A function $\varphi:M \times N \to P$ is **$R$-bilinear** if it is 'linear in both arguments', i.e., >- For all $m \in M$, the function $n \mapsto \varphi(m,n)$ is an $R$-[[linear map]] $N \to P$ >- For all $n \in N$, the function $m \mapsto \varphi(m,n)$ is an $R$-[[linear map]] $M \to P$. > Thus, if $\varphi: M \times N \to P$ is $R$-bilinear, then for all $m \in M$, $n_{1},n_{2} \in N, r_{1},r_{2} \in R$, $\varphi(m, r_{1}n_{1}+r_{2}n_{2})=r_{1}\varphi_{1}(m, n_{1})+ r_{2} \varphi(m, n_{2}),$and similarly for $\varphi(\_{, n})$. > In the special case $P=R$ we call $\varphi$ a **bilinear form**. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```