In the proof that in a PID, every non-zero non-unit is the product of irreducibles...

Question

In proving that all non-zero non-units of a PID are a product of irreducibles, theres:

"We now show that $a$ is a product of irreducibles. If $a$ is irreducible, we are done. Otherwise let $p_1$ be an irreducible such that $p_1|a$. Then $a=p_1c_1$. If $c_1$ is a unit, we are done. "

How did they get that they're done?

I've got $aR = p_1 c_1 R = p_1 R$ if $c_1$ is a unit.

Thanks.

Answer 1

If $c_1$ is a unit, then $a=p_1c_1$ is in fact irreducible. Since we assumed that was not the case, then $c_1$ is not a unit.

I suspect that what they meant to say is "If $c_1$ is irreducible, we are done."