---- > [!theorem] Theorem. ([[characterization of quotienting a module]]) > Let $R$ be a [[ring]] and $M$ an $R$-[[module]]. A [[quotient set|set-quotient]] of $M$ can be endowed with [[linear map|module structure]] such that the natural projection is a [[linear map]] if and only if its elements are [[coset|cosets]] of a [[submodule]] $N \subset M$. In this case, the $R$-[[module]] structure is uniquely determined, with $M / N$ the underlying [[quotient group|(quotient) group]] and [[module|ring action]] $(r, m + N) \mapsto rm+N.$ The following [[universal property]] extends cleanly from its [[characterization of quotienting a group|group analogue]]: > > >[!theorem] Theorem. (Universal property of quotient modules) > Let $N$ be a [[submodule]] of an $R$-[[module]] $M$. Let $\varphi: M \to P$ be a [[linear map]] such that $N \subset \ker \varphi$. > > > >Then there exists a unique $R$-[[linear map]] $\overline{\varphi}:M / N \to P$ so that the diagram > > > >```tikz > >\usepackage{tikz-cd} > >\usepackage{amsmath} > >\begin{document} > >% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFkQBfU9TXfIRRkAjFVqMWbdgHoAct14gM2PASIjy4+s1aIQABW7iYUAObwioAGYAnCAFskZEDghIATNR1T9AHT96WzQACyxFG3snRBc3JE0JXTYAtHDqBjoAIxgGA341IRBbLDMQnAiQO0d46jjEL0TfEACYAA8sOBw4AAIAQm6AiBoYWwYsMBhgAKDQrC4QdKycvNVBNmLS8q4KLiA > >\begin{tikzcd} > >M \arrow[r, "\varphi"] \arrow[d, "\pi"'] & P \\ > >M/N \arrow[ru, "\exists ! \overline{\varphi}"'] & > >\end{tikzcd} > >\end{document} > >``` > >commutes. ^theorem > [!proof]- Proof. ([[characterization of quotienting a module]]) > A [[linear map]] is in particular a [[group homomorphism|homomorphism]] of [[abelian group|abelian groups]], and so the first step towards realizing the natural projection $\pi$ as a [[linear map]] is to note that the [[characterization of quotienting a group]] immediately forces the [[quotient group|quotient]] to be the [[coset|cosets]] $M / N$, where $N$ is a ([[normal subgroup|automatically normal]]) [[subgroup]] of $M$, with [[group|group operation]] $aN + bN=(a+b)N$, and $\pi$ maps $m \mapsto m+N$. Thus it is just a matter of checking how the [[module|action]] of $R$ on the [[group]] $M / N$ should be defined. On the one hand, by definition $\pi(rm)=rm+N.$ On the other hand, if $\pi$ is to be a [[linear map]] then necessarily $\pi(rm)=r \pi (m)=r\big( m + N \big)=rm+rN,$so we need $rN=N$, i.e., $N$ must be a [[submodule]] of $M$, and we need to define the action of $R$ on $M /N$ as $(r,m + N) \mapsto rm+N$. > The question then becomes, is this sufficient? *Does* said action endow $M / N$ with a [[module]] structure? Yes, and to check the action axioms are satisfied is trivial. > As for the [[universal property]], there is not much to check. Existence and uniqueness of $\overline{\varphi}$ as a [[group homomorphism]] is already given in [[characterization of quotienting a group]]; specifically, it is (well-)defined as $\overline{\varphi}(m+N):=\varphi(m)$. We only have to check that it respects scalar multiplication: $\overline{\varphi}\big( r(m+N) \big)=\overline{\varphi}(rm + rN)=\overline{\varphi}(rm+N)=\varphi(rm)=r \varphi(m)=r\overline{\varphi}(m + N),$ as required. ---- #### ----- #### 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 > ```