Prime ideals and quotient rings

#ring_theory

Let $R$ be a commutative ring with unity, let $I \subseteq R$ be an ideal, and let $π : R \to R / I$ be the natural projection homomorphism.

Show that if $℘$ is a prime ideal of $R / I$ , then $π^{- 1} (℘)$ is a prime ideal of $R$ .
Show that the assignment $℘ \mapsto π^{- 1} (℘)$ is injective on the set of prime ideals of $R / I$ .

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L A T E X

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Let $R$ be a commutative ring with unity, let $I\subseteq R$ be an ideal, and let $\pi:R\to R/I$ be the natural projection homomorphism.
\begin{enumerate}[label=\alph*)]
	\item Show that if $\wp$ is a prime ideal of $R/I$, then $\pi^{-1}(\wp)$ is a prime ideal of $R$.
	\item Show that the assignment $\wp\mapsto\pi^{-1}(\wp)$ is injective on the set of prime ideals of $R/I$.
\end{enumerate}