Quotients and direct products
Let
- Let
be ideals, and put . Use the First Isomorphism Theorem to prove that . - Prove the prime ideals of
have the form where is a prime ideal for . (Omit the proof that this is an ideal.)
View code
Let $R_1,\ldots, R_k$ be commutative rings, and set $R=R_1\times \cdots \times R_k$.
\begin{enumerate}[label=\alph*)]
\item Let $I_j\subset R_j$ be ideals, and put $I=I_1\times \cdots \times I_k$. Use the First Isomorphism Theorem to prove that $R/I\simeq R_1/I_1\times \cdots \times R_k/I_k$.
\item Prove the prime ideals of $R$ have the form $R_1\times \cdots \times R_{j-1}\times P_j\times R_{j+1}\times \cdots \times R_k$ where $P_j\subset R_j$ is a prime ideal for $1\leq j\leq k$. (Omit the proof that this is an ideal.)
\end{enumerate}