Quotients and direct products

#ring_theory

Let $R_{1}, \dots, R_{k}$ be commutative rings, and set $R = R_{1} \times \dots \times R_{k}$ .
(a) Let $I_{j} \subset R_{j}$ be ideals, and put $I = I_{1} \times \dots \times I_{k}$ . Use the First Isomorphism Theorem to prove that $R / I ≃ R_{1} / I_{1} \times \dots \times R_{k} / I_{k}$ .
(b) Prove the prime ideals of $R$ have the form $R_{1} \times \dots \times R_{j - 1} \times P_{j} \times R_{j + 1} \times \dots \times R_{k}$ where $P_{j} \subset R_{j}$ is a prime ideal for $1 \leq j \leq k$ . (Omit the proof that this is an ideal.)

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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}