# Tagged: principal ideal domain

## Problem 724

Let $R$ be a principal ideal domain. Let $a\in R$ be a nonzero, non-unit element. Show that the following are equivalent.

(1) The ideal $(a)$ generated by $a$ is maximal.

(2) The ideal $(a)$ is prime.

(3) The element $a$ is irreducible.

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## Problem 535

**(a)** Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal.

**(b)** Prove that a quotient ring of a PID by a prime ideal is a PID.

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Ring theory

by
Yu
· Published 08/08/2017
· Last modified 08/09/2017

## Problem 534

Let $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$.

Prove that the quotient ring $\Z[i]/I$ is finite.

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Ring theory

by
Yu
· Published 12/21/2016
· Last modified 08/12/2017

## Problem 228

Let $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.

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Ring theory

by
Yu
· Published 11/11/2016
· Last modified 08/01/2017

## 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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