Question

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.

Short answer

It is proved that$p$ is irreducible.

Step-by-step solution

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 $R$.

Proving that p is irreducible

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.