---- > [!theorem] Theorem. ([[characterization of quotienting a ring]]) > Let $R$ be a [[ring]]. A [[quotient set|set-quotient]] of $R$ can be endowed with [[ring]] structure such that the natural projection is a [[ring homomorphism|ring homomorphism]] if and only if its elements are [[coset|cosets]] of an [[ideal]] $I \subset R$. In this case, the [[ring]] structure is uniquely determined, with $R / I$ the underlying [[quotient group|(quotient) group]] and multiplication given by $(a+I)(b+I):= ab+I.$ > The following [[universal property]] extends cleanly from its [[characterization of quotienting a group|group analog]]. > >[!theorem] > > > > > >Let $R$ be a [[ring]] and $I \subset R$ an [[ideal]]. Then for any [[ring homomorphism|ring homomorphism]] $\varphi: R \to S$ satisfying $I \subset \text{ker }\varphi$, there is a unique [[ring homomorphism]] $\overline{\varphi}$ so that the diagram > >```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACUQBfU9TXfIRRkAjFVqMWbdgHoAkt14gM2PASIjy4+s1aIQAZW7iYUAObwioAGYAnCAFskZEDghIATNR1T9AHT96WzQACyxFG3snRBc3JE0JXTYAtHDqHDosBjZIMFZqBiw8tjgIQqgQahCYOgr9XPyQBjoAIxgGAAV+NSEQWywzEJwIkDtHePT3RC9E3xAAmAAPLDgcOAACAEJ1gIgaGFtCvOAAoNCsLkqm1vau1UE2fsHhrgouIA > \begin{tikzcd} > R \arrow[r, "\varphi"] \arrow[d, "\pi"', no head] & S \\ > R/I \arrow[ru, "\exists ! \overline{\varphi}"'] & > \end{tikzcd} > \end{document} > >``` > commutes. > > > In particular, $\overline{\varphi}$ is defined $\overline{\varphi}(r+I):=\varphi(r)$. ^theorem > [!note] Remark. > To be precise, the [[universal property]] above is claiming that $R \xrightarrow{\pi}R / I$ is [[terminal object|initial]] in the [[subcategory]] of the [[coslice category]] $\mathsf{Ring}^{R}$ obtained by keeping just those [[ring homomorphism|ring homomorphisms]] having $I$ in their [[kernel of a ring homomorphism|kernel]]. ^note > [!proof]- Proof. ([[characterization of quotienting a ring]]) > At the very least, $R$ is an ([[abelian group|abelian]]) [[group]] under addition, and any quotient of $R$ must be therefore an ([[abelian group|abelian]]) [[group]] under addition. The [[characterization of quotienting a group]] tells us that there is only one way for a [[quotient set|(set-)quotient]] of $R$ to have [[group]] structure: if we take as elements the [[coset|cosets]] of [[normal subgroup]] $I$ (well, $R$ is [[abelian group|abelian]], so any [[subgroup]] of it is [[normal subgroup|normal]]) and define addition as $(r_{1} + I) + (r_{2} + I)=(r_{1}+r_{2}) + I.$ Thus, our task is to see how we might endow the [[abelian group]] [^1] $R / I$ with multiplication and identity to make it into a [[ring]] and $\pi: R \to R / I$ into a [[ring homomorphism]]. Since $\pi(ab)=ab+I$ by the definition of $\pi$ as a [[group homomorphism]], we are forced in our attempts to define multiplication $(a+I)$, $(b+I)$ in $R / I$ by the fact $(a+I)(b+I)=\pi(a)\pi(b)=\pi(ab)=ab+I.$ *If* this operation is [[well-defined]], then it indeed makes $R / I$ into a [[ring]]: associativity is inherited and the [[coset]] $1+I$ acts as identity. > Our task now is to find necessary and sufficient conditions which make the [[binary operation|operation]] [[well-defined]]. An easy necessary condition is that by [[kernel iff normal subgroup]] and the definition of [[kernel of a ring homomorphism]] (namely, the kernel of RH is just kernel of underlying GH), $I$ is the kernel of $\pi:R \to R / I$. So necessarily $I$ must be an [[ideal]]. > We claim this is sufficient. To see this, let $I$ be an [[ideal]] and $a'+I=a''+I$ and $b'+I=b''+I$. This means $a''-a'\in I$ and $b''-b' \in I$, then $a''b'' - a'b'=a''b'' - a''b'+a''b'-a'b'=a''(a''-b')+(a''-a')b' \in I$ which says precisely that $a'b'+I=a''b'' +I$. > Finally, we prove the [[universal property]], though there is not much to be said. The [[characterization of quotienting a group|universal property of quotient groups]] tells us that $\overline{\varphi}$ already exists and is unique as a [[group homomorphism]]. So it just has to be checked that it is a [[ring homomorphism]]. Identity is respected since $\varphi(1+I)=\varphi(1)=1_{S}$, and multiplication is respected since $\begin{align} \overline{\varphi}\big( (r_{1} +I) (r_{2} + I) \big)= & \varphi\big( r_{1} r_{2} \big) \\ = & \varphi(r_{1}) \varphi(r_{2}) \\ = & \overline{\varphi}(r_{1} + I) \overline{\varphi}(r_{2} + I). \end{align}$ ---- #### [^1]: It is automatically [[abelian group|abelian]] [[group homomorphisms preserve structure ]] as the image of the abelian group $R$ under the [[kernel iff normal subgroup|natural projection homomorphism]] $\pi$, see [[group homomorphisms preserve structure]]. ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as 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 > ```