---- > [!definition] Definition. ([[cokernel of a module homomorphism]]) > [[categorical cokernel|Cokernels]] exist in $R$-$\mathsf{Mod}$. In particular, given $R$-[[module|modules]] $M,N,P$ and an $R$-[[module]] [[linear map|homomorphism]] $\varphi: M \to N$, the [[quotient module]] $\frac{N}{\im \varphi}$ is [[terminal object|initial]] with respect to the property of factoring $R$-[[linear map|module homomorphisms]] $\beta:N \to P$ satisfying $\beta \circ \varphi=0$. So $\text{coker }\varphi \cong \frac{N}{\im \varphi}$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFkQBfU9TXfIRQBGclVqMWbAHLdeIDNjwEiAJjHV6zVohAAFOXyWCio4eK1TdAHWs4YADxzAAxhADWMAE4ACLrfovNAALLG5xGCgAc3giUAAzLwgAWyQyEBwIJFEJbTYAuiDQkGoGOgAjGAY9fmUhEC8sKOCcQxBElOzqTKQAZk1JHRBbNDDSiqqa4xVdRubWngSk1MQcnsR1XKth60qcOhKQMsrq2pNZppa2jpX+jKyN0qwwIag6OGDIw8sh20csOA4OA+ACEPlsEBo3gYzxgwFsezoFC4h2OkzOMwalwW8huaW6D02lTAUD66R+bGI4S4QA > \begin{tikzcd} > M \arrow[r, "\varphi"'] \arrow[rr, "0", bend left] & N \arrow[d, "\pi"'] \arrow[r, "\beta"'] & P \\ > & \text{coker }\varphi \arrow[ru, "{\exists ! \overline{\beta}}"', dashed] & > \end{tikzcd} > \end{document} > ``` ^definition > [!justification] > Same deal as in the [[cokernel of an abelian group homomorphism|case with]] [[abelian group|abelian groups]], just this time invoke the [[characterization of quotienting a module|universal property of quotient modules]] instead of the [[characterization of quotienting a group|universal property of quotient groups]]. ^justification ---- #### ---- #### 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 > ```