----- > [!proposition] Proposition. ([[passing ideal to polynomial ring]]) > Let $R$ be a [[commutative ring]]. An [[ideal]] $I \subset R$ generates an [[ideal]] of the [[polynomial 4|polynomial ring]] $R[x]$: $IR[x]:=\{ a_{0} + a_{1}x + \dots + a_{d}x^{d} \in R[x] : a_{i} \in I \text{ for all }i\}.$ > > Note that $IR[x]$ is [[prime ideal|principal]] if $I$ is [[prime ideal|principal]]. > Moreover, $\frac{R[x]}{IR[x]} \cong \frac{R}{I}[x]$ > from which we can also see that $IR[x]$ is [[prime ideal|prime]] in $R[x]$ if $I$ is [[prime ideal|prime]] in $R$. > [!proof]- Proof. ([[passing ideal to polynomial ring]]) > Define a [[surjection]] $\varphi:R[x]\to \frac{R}{I}[x]$ by $\varphi\left( \sum_{\ell}a_{\ell}x^{\ell} \right):= \sum_{\ell}(a_{\ell}+I)x^{\ell}.$ > The RHS is zero iff each $a_{\ell}+I$ is zero in $\frac{R}{I}$, that is, iff each $a_{\ell} \in I$. Thus, $\ker \varphi=\left\{ \sum_{\ell} a_{\ell} x^{\ell} : a_{\ell} \in I \text{ for all }\ell \right\}.$ > This is precisely the definition of $IR[x]$. The result now follows from the [[first isomorphism theorem for rings]]. ----- #### ---- #### 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 > ```