### Algebra I: Commutative Rings. Rephrase Universal property of rings

parent 42d5428d
 ... @@ -332,17 +332,18 @@ ... @@ -332,17 +332,18 @@ \textbf{isomorphism}. \textbf{isomorphism}. \end{definition} \end{definition} \begin{theorem} \begin{theorem}[Universal property of Rings] Let $R$ and $S$ be commutative rings, and let $\varphi: R \to S$ be a Let $R$ and $S$ be commutative rings, and let $\varphi \colon R \to S$ be a homomorphism. If $s_1, \ldots, s_n \in S$, then there exists a unique homomorphism. For fixed $s_1, \ldots, s_n \in S$, there exists a unique homomorphism homomorphism % % \begin{align*} \begin{align*} \Phi: R[x_1, \ldots, x_n] \to S \Phi \colon R[x_1, \ldots, x_n] & \to S \\ x_i & \mapsto s_i \end{align*} \end{align*} % % with $\Phi(x_i) = s_i$ for all $i$ and $\Phi(r) = \varphi(r)$ for all $r \in with$\Phi(r) = \varphi(r)$for all$r \in R$. That is,$\Phi$uniquely R$. extends $\varphi$ from $R$ to $R[x_1, \ldots, x_n]$, sending $x_i$ to $s_i$. \end{theorem} \end{theorem} \begin{definition}[Evaluation map] \begin{definition}[Evaluation map] ... ...
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