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Answers without the blur. Sign up and see all textbooks for free! Q10.1-23E

Expert-verified Found in: Page 331 ### Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624 # Let p be nonzero, non-unit element of R such that whenever ${\mathbit{p}}{\mathbf{/}}{\mathbit{c}}{\mathbit{d}}$ , then ${\mathbf{p}}{\mathbf{/}}{\mathbf{c}}$ or ${\mathbf{p}}{\mathbf{/}}{\mathbf{d}}$. Prove that ${\mathbit{p}}$ is irreducible.

It is proved that $p$ is irreducible.

See the step by step solution

## Step 1: Irreducible element

A nonzero element ${p}{\in }{R}$ is said to be irreducible provided that p is not a unit and the only divisor of p are its associates and the units of R.

Now, Let $p$ be nonzero, non-unit element of $R$ such that whenever $p/cd$, then $p/c$ or $p/d$.

Therefore, $p$ is prime element in $R$.

## Step 2: Proving that p is irreducible

Now,$p/c⇒c=ap$

Now, $p/cd⇒p=cd⇒p=apd⇒p\left(1-ad\right)=0$

$⇒ad=1$

Therefore, $d$ is unit element

Hence $p$ is irreducible. ### Want to see more solutions like these? 