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## Homework Statement

Let ##R## be a PID and let ##\pi\in{R}## be an irreducible element. If ##B\in{R}## and ##\pi\not{|}B##, prove ##\pi## and ##B## are relatively prime.

## Homework Equations

##\pi## being irreducible means for any ##a,b\in{R}## such that ##ab=\pi##, one of #a# and #b# must be a unit (meaning they have multiplicative inverses).

##\pi## and ##B## being relatively prime means their GCD is the identity element of R (call it 1).

My book also has a theorem saying if ##\alpha## is an irreducible element in a PID and ##\alpha|xy## then ##\alpha|x## or ##\alpha|y##.

The reason I am wary to use this is because they cited the problem I'm currently asking about in their proof...

## The Attempt at a Solution

EDIT EDIT EDIT: What I had below was wrong (see post 7 if you want to see what I had and why it's wrong).

So I'm still very much stuck :(

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