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Q10.1-23E

Expert-verifiedFound in: Page 331

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 ${\mathit{p}}{\mathbf{/}}{\mathit{c}}{\mathit{d}}$ , then ${\mathbf{p}}{\mathbf{/}}{\mathbf{c}}$ or ${\mathbf{p}}{\mathbf{/}}{\mathbf{d}}$. Prove that ${\mathit{p}}$ is irreducible.**

It is proved that $p$ is irreducible.

**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 $R$.

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

Now, $p/cd\Rightarrow p=cd\Rightarrow p=apd\Rightarrow p(1-ad)=0$

$\Rightarrow ad=1$

Therefore, $d$ is unit element

Hence $p$ is irreducible.

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