## In a Principal Ideal Domain (PID), a Prime Ideal is a Maximal Ideal

## Problem 175

Let $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$.

Show that $P$ is a maximal ideal in $R$.

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Let $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$.

Show that $P$ is a maximal ideal in $R$.

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Let $R$ be a commutative ring and let $P$ be an ideal of $R$. Prove that the following statements are equivalent:

**(a)** The ideal $P$ is a prime ideal.

**(b)** For any two ideals $I$ and $J$, if $IJ \subset P$ then we have either $I \subset P$ or $J \subset P$.

Let $R$ be a commutative ring. An ideal $I$ of $R$ is said to be **irreducible** if it cannot be written as an intersection of two ideals of $R$ which are strictly larger than $I$.

Prove that if $\frakp$ is a prime ideal of the commutative ring $R$, then $\frakp$ is irreducible.

Add to solve later Let $R$ be a commutative ring with $1 \neq 0$.

An element $a\in R$ is called **nilpotent** if $a^n=0$ for some positive integer $n$.

Then prove that if $a$ is a nilpotent element of $R$, then $1-ab$ is a unit for all $b \in R$.

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