Im trying to learn the proof of Every Principal Ideal Domain is Unique Factorization Domain

First we take `a E D`(a is non zero,non unit) `a=p1.p2...pn` and `a=q1.q2....qs`

Then its enough to prove the factorization is unique


Then the proof says that

p1 divides LHS Which implies p1 divides RHS.

How can we say that?

Then it says that

Without Loss of generality  q1=U1p1(For some unit U1 E D)

How can we say this also?

  • $\begingroup$ In a PID, the irreducibles $p_i, q_i$ are also prime. Thus $p_1 | q_1 \dots q_n$ implies $p_1$ divides one of them. Hence the $p_1 = u_1 q_i$ (for some $i$ which is assumed 1 here). $\endgroup$ – Weaam Oct 20 '15 at 5:08
  • 1
    $\begingroup$ There is no reason to assume from the start that both factorizations have $n$ terms. $p_1$ divides LHS too because LHS = RHS, $\endgroup$ – steven gregory Oct 20 '15 at 5:10
  • $\begingroup$ @StevenGregory That was a typing mistake.Corrected it. $\endgroup$ – techno Oct 20 '15 at 5:12

You know that in a PID, a prime number is the same as an irreducible. Thus if $p_1$ divides the RHS of the equality it divides some $q_i$ and without loss of generality (because in a PID the product is associative and commutative) we can assume $q_1=q_i.$ It follows that since $q_1$ is a prime then $p_1$ is an associate of $q_1.$


You could also use Kaplansky's theorem: an integral domain $D$ is a UFD if and only if every non-zero prime ideal contains a non-zero prime element. http://sierra.nmsu.edu/morandi/oldwebpages/Math581Fall2012/Kaplansky.pdf

If $P$ is a non-zero prime ideal, it is principal since all ideals are principal in a PID and thus $P=(p)$ for some non-zero $p$. Moreover, $p$ is prime. So it is a UFD.

Would probably really freak your professor out.

  • $\begingroup$ Most probably the person correcting the answer paper will not know of this.I dont want to take my chances :) lol $\endgroup$ – techno Oct 20 '15 at 8:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.