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Recall that an integral domain $R$ is atomic if every nonzero nonunit admits at least one factorization into irreducible elements. (Indeed, hard-core factorization theorists have replaced the word "irreducible" by "atom".)

From prior reading, I happen to know that there exist atomic integral domains $R$ such that the univariate polynomial ring $R[t]$ is not atomic. This is a somewhat surprising pathology, because the implication is true if both instances of "atomic" are replaced by "UFD", "Noetherian" or "Ascending Chain Condition on Principal Ideals".

But I don't know a precise example or a reference, and I would like one for an expository article I'm writing. Of course, the chronologically earlier and logically simpler the example, the better.

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  • $\begingroup$ +1, but I removed the LaTeX from the title, as it served no purpose, as far as I can tell. $\endgroup$
    – Harry Gindi
    Commented May 19, 2010 at 9:06

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According to the book "Non-Noetherian commutative ring theory" by S.T. Chapman and S. Glaz the question was first asked in "Factorization of integral domains" by D.D. Anderson, D.F. Anderson, M. Zafrullah, Journal of Pure and Applied Algebra 69 (1990) 1-19 (question 1). An answer was given here by M. Roitman.

There it was conjectured that $R[X]$ atomic $\implies$ $R[X,Y]$ is also atomic.

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  • $\begingroup$ Thanks, Gjergji. I will acknowledge you in my article. Since it is expository, there is no guarantee that it will see the light of day -- or rather, traditional publication -- but it will be internet available, at least. $\endgroup$
    – Pete L. Clark
    Commented May 19, 2010 at 9:49
  • $\begingroup$ Thanks! Your articles are really nice, may I ask what this next one will be about? $\endgroup$
    – Gjergji Zaimi
    Commented May 19, 2010 at 10:15
  • $\begingroup$ I'm working on revising / slightly expanding math.uga.edu/~pete/factorization.pdf, which was first written about a year ago. $\endgroup$
    – Pete L. Clark
    Commented May 19, 2010 at 11:21
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    $\begingroup$ Here's a link to an updated version of the factorization notes in the above comment, which include a reference to Roitman's paper. $\endgroup$
    – Arrow
    Commented May 5, 2020 at 10:22
  • $\begingroup$ The link to sciencedirect.com is broken. I'm also unable to find any snapshot saved on the Wayback Machine. $\endgroup$
    – The Amplitwist
    Commented May 15, 2023 at 5:09

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