---- $R$ is a [[ring]]. > [!definition] Definition. ([[kernel of a module homomorphism]]) > The **kernel** of a [[linear map|homomorphism]] $\varphi:M \to N$ of $R$-[[module|modules]] is the [[kernel of a group homomorphism|kernel]] of underlying [[group homomorphism]]:[^1] $\ker \varphi:= \{ m \in M: \varphi(m)=0_{N} \}.$ > > This is a 'kernel' in the [[categorical kernel|categorical sense]]: $\ker \varphi$ is [[terminal object|final]] with respect to the property of factoring $R$-[[module]] [[linear map|homomorphisms]] $\alpha:M \to N$ satisfying $\varphi \circ \alpha=0$: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFkQBfU9TXfIRQBGclVqMWbAHLdeIDNjwEiAJjHV6zVohAAFOXyWCio4eK1TdAHWs4YADxzAA1jABOAAi6367tAAWWNziMFAA5vBEoABm7hAAtkhkIDgQSKIgDHQARjAMevzKQiDuWOEBOCCakjogtoyBdIYgcYkZ1GlI6lm5+YXGKrplFVU12my+dP5BLW1JiADMnemImTh0WAxsARAQLtW9eQVFJsPllXPxCyldS+NW9daOWHA4cJ4AhJ62EDQeDCwYBgwAaDCaXEO2WOAwEQ1KFyqPFi12SK261DyYCgSEWKUsdWIIS4QA > \begin{tikzcd} > M \arrow[r, "\alpha"'] \arrow[rd, "\exists ! \overline{\alpha}"'] \arrow[rr, "0", bend left] & N \arrow[r, "\varphi"'] & P \\ > & \text{ker }\varphi \arrow[u, hook] & > \end{tikzcd} > \end{document} > ``` > ^definition > [!justification] > Truly immediate. We need to show $\ker \varphi$ satisfies the [[universal property]] of [[categorical kernel|kernels in]] $R$-$\mathsf{Mod}$. But this is easy: from [[universal property of group homomorphism kernels]] (which also is easy) we know that $\overline{\alpha}$ exists as just $\alpha$ itself, and by hypothesis $\alpha$ is a [[linear map]]. That is all! ^justification [^1]: Evidently $\ker \varphi$ is a [[submodule]] of $M$: it is known to be an [[abelian group|abelian subgroup]], and stability under the $R$-action follows from $\varphi(m)=0 \implies \varphi(r \cdot m)=r \varphi(m)=0.$ ---- #### ---- #### 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 > ```