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I am looking for an example of a commutative ring with $1$, in which every ideal is generated by a single element, for which the conclusion of the structure theorem for finitely generated modules is wrong. (The ring is allowed to have zero divisors, so it is not a PID).

Are there any examples? What happens if one drops the other conditions instead (commutativity, $1\in R$)? Does then the structure theorem still fail ?

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    $\begingroup$ If you don't have 1 in R, then all hell breaks loose. For example, you can take any abelian group $A$ and let $R$ act on it by $r⋅a=0\;\forall r\in R$ and $a\in A$. This is certainly not a direct sum of quotients of the regular module $\endgroup$
    – Alex B.
    Commented Nov 16, 2010 at 15:07

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The note linked to in Timothy Wagner's answer has been replaced by another one, which only shows the structure theorem for PIDs, so it may be worth to point out that the structure theorem holds for any principal ideal ring (PIR), possibly with zero divisors. Namely, a theorem of Zariski-Samuel tells us that a PIR is a direct product of PIDs and local artinian PIRs. For these, the structure theorem holds and one has uniqueness (for the latter, see Keenan Kidwell's question he mentioned in the comment). Since a module $V$ over a ring $R= R_1 \times \dotsb \times R_n$ decomposes canonically as $V= Ve_1\oplus \dots Ve_n$, where the $e_i$ are the obvious idempotents, and $Ve_i$ is an $R_i$-module, we are done.

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I am unable to write this is in comments. While this is not an answer to your question, a similar structure theorem holds for Principal ideal rings where every finitely generated module is isomorphic to a direct sum of cyclic modules.

http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Principal.ideals/principal.ideals.070702.pdf

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  • $\begingroup$ the upper PDF only claims/shows the existence part of the structure theorem. So I guess one can look for a Principal ideal ring with zero divisors, where the uniqueness part fails. $\endgroup$
    – HenrikRüping
    Commented Nov 16, 2010 at 14:46
  • $\begingroup$ According to theorems 7.3,7.5 in Lang's Algebra, the decomposition is unique. $\endgroup$
    – Timothy Wagner
    Commented Nov 16, 2010 at 15:16
  • $\begingroup$ A related question I asked a while back (where somebody also linked to the above pdf) is mathoverflow.net/questions/22722/… For principal artin local rings (like $\mathbb{Z}/p^n\mathbb{Z}$ for a prime $p$ and $n\geq 1$), every element is either a unit or nilpotent, and the structure theorem holds and one has uniqueness. $\endgroup$
    – Keenan Kidwell
    Commented Nov 16, 2010 at 16:06
  • $\begingroup$ Sorry, Pierre, but I don't understand f(x, y) divides f(x, v) in the proof of Proposition 2. What does "maximal" mean - bigger than all, or not smaller than any? And something (which I would call experience, had I any in this field of algebra) makes me seriously doubt that a proof of this result fits in a page... $\endgroup$
    – darij grinberg
    Commented Nov 17, 2010 at 15:08
  • $\begingroup$ The link seems to be dead - and I found it neither in the Wayback Machine nor on the Pierre-Yves Gaillard's new site. Perhaps somebody else might have better luck finding it... $\endgroup$
    – Martin Sleziak
    Commented Aug 14, 2022 at 9:10

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