---- > [!definition] Definition. ([[tensor functor]]) > Let $R$ be a [[ring]] and $N$ an $R$-[[module]]. Then the assignment $M \mapsto M \otimes_{R} N$ gives rise to a [[covariant functor]] $R$-$\mathsf{Mod} \to R$-$\mathsf{Mod}$, called the **[[tensor product of modules|tensor]] functor** and denoted by $\_ \otimes_{R} N$ or $T_{N}$. It behaves on [[linear map|morphisms]] $\alpha:M_{1} \to M_{2}$ as $T_{N}(\alpha)(m \otimes n) := \alpha(n) \otimes n,$ in other words, as the [[tensor product of module homomorphisms]] $\alpha \otimes \id_{N}$. Similarly, one may define a [[covariant functor|functor]] $M \otimes_{R} \_:R\text{-}\mathsf{Mod} \to R\text{-}\mathsf{Mod}.$ > > There of course is also a [[bifunctor]] $\_ \otimes_{R} \_$ taking a pair of $R$-[[module|modules]] $M,N$ to their [[tensor product of modules|tensor product]] $M \otimes_{R} N$ and a pair of morphisms $f:M \to M'$, $g:N \to N'$ to the **tensor product of $R$-linear maps** $\begin{align} f \otimes g: M \otimes_{R} N &\to M' \otimes_{R} N' \\ (f \otimes g)(m \otimes n) &\mapsto f(m) \otimes_{R} g(n). \end{align}$ - [ ] [[Kronecker product]] of matrices > [!basicproperties] > - [[tensoring preserves isomorphisms and surjectivity, but perhaps not injectivity]] > - $(f \otimes g) \circ (h \otimes i)=(f \circ h) \otimes (g \circ i)$ ^properties ---- #### ---- #### 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 > ```