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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.

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1 Answer 1

up vote 3 down vote accepted

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

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Excellent answer. In acronyms : every PID is a UFD. –  Georges Elencwajg Apr 27 '13 at 8:06
    
@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? –  ramanujan_dirac Apr 27 '13 at 9:11
    
@Georges: We don't need to use this fact. From the ideal structure one can immediately check UFD, with the only prime element $t$. –  Martin Brandenburg Apr 27 '13 at 9:11
    
@MartinBrandenburg: How can you say t is the only prime element? –  ramanujan_dirac Apr 27 '13 at 9:21
    
@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. –  egreg Apr 27 '13 at 9:23

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