---- > [!definition] Definition. ([[tensor product of module homomorphisms]]) Let $R$ be a [[commutative ring|(say, commutative)]] [[ring]] and $M,N, M',N'$ be $R$-[[module|modules]]. For $R$-[[linear map|linear maps]] $f:M \to M'$, $g:N \to N'$, there exists a unique $R$-linear map $f \otimes g:M \otimes N \to M' \otimes N'$ such that $(f \otimes g)(m \otimes n)=f(m) \otimes g(n),$ called the **tensor product of $f$ and $g$**. ^definition > [!justification] > Uniqueness is given by assumption. Existence follows from the [[universal property]] for the [[tensor product of modules]] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFkACAHW7wFt4nAHIgAvqXSZc+QigCM5KrUYs27AOQ9uEAfAD6AJREbxkkBmx4CRMvOX1mrRB226sguEZHjlMKADm8ESgAGYAThD8SGQgOBBIAEzUjmouvO6eINQMdABGMAwACtLWciDhWAEAFjhmYZHRiLHxSIoqTmy89OFo1Vj1IBFRSdStiO2pziC8MAAeWHA4cJwAhG40MOEMWGAwwN10vf1i2SC5BcWlsmyVNXViFGJAA > \begin{tikzcd} > M \times N \arrow[d, "\otimes"'] \arrow[r, "\varphi"] & M' \otimes_R N' \\ > M \otimes_R N \arrow[ru, "\exists ! \overline{\varphi}"'] & > \end{tikzcd} > \end{document} > ``` > by defining $\varphi$ in the pictured diagram by $\varphi(m,n):=f(m) \otimes g(n)$. Then commutativity yields a unique $\overline{\varphi}:M \otimes_{R} N \to M' \otimes_{R} N'$ satisfying $\overline{\varphi}(m \otimes n)=f(m) \otimes g(n)$. We therefore have found that $\overline{\varphi}=f \otimes g$. ---- #### [[Kronecker product]] ---- #### 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 > ```