1. ## Irreduciple element -problem

I have this kind of exercise in which I need help:

Let R be integral domain. Assume that there is an element $\displaystyle p \in R\setminus U(R), p \neq 0$, so that if $\displaystyle p\mid xy$ then $\displaystyle p\mid x$ or $\displaystyle p\mid y$ with every $\displaystyle x,y \in R$. Show that $\displaystyle p$ is irreducible. (U(R) = the group of units of R)

Any help would be great.

2. Originally Posted by Ester
I have this kind of exercise in which I need help:

Let R be integral domain. Assume that there is an element $\displaystyle p \in R\setminus U(R), p \neq 0$, so that if $\displaystyle p\mid xy$ then $\displaystyle p\mid x$ or $\displaystyle p\mid y$ with every $\displaystyle x,y \in R$. Show that $\displaystyle p$ is irreducible. (U(R) = the group of units of R)

Any help would be great.

Suppose $\displaystyle p=ab\Longrightarrow p\mid ab\Longrightarrow p\mid a\,\,or\,\,p\mid b$ .

WLTG suppose $\displaystyle p\mid a\Longrightarrow a=px\Longrightarrow p = ab=pxb\Longrightarrow xb=1$.

End now the proof.

Tonio