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In Matsumura's ''Commutative ring theory'' he proves the folowing: If $A$ is a regular UFD, the ring of formal series $A[\![ X ]\!]$ is a UFD (page $165$). Just below he says that we can't drop regularity from the hypothesis, without giving any counterexample. Is there a non-insane example of UFD $A$ such that $A[\![X]\!]$ is not?

Being a bit optimistic, is true that for every non-regular UFD $A$ the ring $A[\![X]\!]$ is not a UFD?

J. W. Tanner
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feriado
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1 Answers1

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The wiki page on UFDs offers this example:

Let $R$ be $k[x,y,z]/(x^2+y^3+z^7)$ localized at $(x,y,z)$. Then $R$ is a UFD but $R[[X]]$ is not.

Update: looking through the links that we've found since I posted this, there's a reference to

Samuel, Pierre. "On unique factorization domains." Illinois Journal of Mathematics 5.1 (1961): 1-17.

Which uses this example but also demands $k$ is a perfect field of characteristic $2$. Even then, the reasoning is quite involved and probably too much for a solution here. Currently the paper is open to all, so it will have to do.

That was the early days of the example, so perhaps someone has found a more streamlined explanation since then.

rschwieb
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