The problem is from Artin.

Prove that the ring $\mathbb{R}[[t]]$ of formal power series given by $p(t)=a_0 + a_1 t+ a_2 t^2 + \cdots$ is an UFD.

I have no idea how to do this. From the couple of things that I know about UFDs is that I could show that every irreducible element is prime, or I could show that every chain of ideals terminates. Any help will be appreciated.


Show that each ideal $\neq 0$ has the form $\mathfrak a = t^k\mathbb R[[t]] $.

  • 1
    $\begingroup$ Excellent answer. In acronyms : every PID is a UFD. $\endgroup$ – Georges Elencwajg Apr 27 '13 at 8:06
  • $\begingroup$ @Hagen: Thanks! Cud you help me with a similar question, where I have to show that the ring of Laurent polynomials is a principal ideal domain? $\endgroup$ – user23238 Apr 27 '13 at 9:11
  • $\begingroup$ @Georges: We don't need to use this fact. From the ideal structure one can immediately check UFD, with the only prime element $t$. $\endgroup$ – Martin Brandenburg Apr 27 '13 at 9:11
  • $\begingroup$ @MartinBrandenburg: How can you say t is the only prime element? $\endgroup$ – user23238 Apr 27 '13 at 9:21
  • $\begingroup$ @ramanujan_dirac Every power series $a_0+a_1t+\dots$ with $a_0\ne0$ is invertible in $\mathbb{R}[[t]]$; this should answer both your two new questions. $\endgroup$ – egreg Apr 27 '13 at 9:23

Your Answer

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