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Q10.1-23E

Expert-verified

Abstract Algebra: An Introduction

Found in: Page 331

Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

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Short Answer

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

It is proved that $p$ is irreducible.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 $p / c d$ , then $p / c$ or $p / d$ .

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

Step 2: Proving that p is irreducible

Now, $p / c \Rightarrow c = a p$

Now, $p / c d \Rightarrow p = c d \Rightarrow p = a p d \Rightarrow p (1 - a d) = 0$

$\Rightarrow a d = 1$

Therefore, $d$ is unit element

Hence $p$ is irreducible.

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