# Tagged: irreducible element

## 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 519

Prove that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).

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Ring theory

by
Yu
· Published 07/24/2017
· Last modified 07/25/2017

## Problem 518

Prove that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD).

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Ring theory

by
Yu
· Published 11/12/2016
· Last modified 08/11/2017

## Problem 177

Let $R$ be a principal ideal domain (PID). Let $a\in R$ be a non-unit irreducible element.

Then show that the ideal $(a)$ generated by the element $a$ is a maximal ideal.

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