I'm learning a proof about UFD's but I'm not following this particular part:
Given a ring A which is a UFD, with a non-zero non-unit element p which is a prime element.
Then if you take an arbitrary product ab from (p) (proper non zero ideal), you can deduce there is c such that ab = pc.
Now the proof states that due to the unique factorisation property, we have that either a or b is divisible by p, that is to say that you can write just one of them as a product of p and something else. Meaning a is in (p) or b is in (p)
I don't understand how we deduced this, surely both of them could have a unique factorisation which has p? i.e c = p(e...f)(g...h) so a = p(e...f) and b = p(g...h)
Almost invariably, in mathematical communication, “Α or B” is understood to include the case where both are true. So, yes, they could both have a unique factorisation involving $p$.