Unverified Commit 84cf7c6e authored by rrueger's avatar rrueger
Browse files

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]
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment