Irreduciple element -problem

**Ester** · February 4th 2011, 03:28 AM

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

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

Any help would be great.

**tonio** · February 4th 2011, 03:43 AM

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 $p \in R\setminus U(R), p \neq 0$ , so that if $p\mid xy$ then $p\mid x$ or $p\mid y$ with every $x,y \in R$ . Show that $p$ is irreducible. (U(R) = the group of units of R)

Any help would be great.

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

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

End now the proof.

Tonio

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