Structure theorem of f.g. modules over a (non) PID - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:17:32Z http://mathoverflow.net/feeds/question/46231 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46231/structure-theorem-of-f-g-modules-over-a-non-pid Structure theorem of f.g. modules over a (non) PID HenrikRüping 2010-11-16T13:37:07Z 2011-09-08T15:24:29Z <p>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).</p> <p>Are there any examples? What happens if one drops the other conditions instead (commutativity, $1\in R$)? Does then the structure theorem still fail ?</p> http://mathoverflow.net/questions/46231/structure-theorem-of-f-g-modules-over-a-non-pid/46238#46238 Answer by Timothy Wagner for Structure theorem of f.g. modules over a (non) PID Timothy Wagner 2010-11-16T14:34:18Z 2010-11-16T14:44:09Z <p>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.</p> <p><a href="http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Principal.ideals/principal.ideals.070702.pdf" rel="nofollow">http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Principal.ideals/principal.ideals.070702.pdf</a></p> http://mathoverflow.net/questions/46231/structure-theorem-of-f-g-modules-over-a-non-pid/74890#74890 Answer by F. Ladisch for Structure theorem of f.g. modules over a (non) PID F. Ladisch 2011-09-08T15:24:29Z 2011-09-08T15:24:29Z <p>The note linked to in Timothy Wagner's answer has been replaced by <a href="http://www.iecn.u-nancy.fr/~gaillapy/DIVERS/PID/pid-110807e.pdf" rel="nofollow">another one</a>, 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 <a href="http://mathoverflow.net/questions/22722/why-are-finitely-generated-modules-over-principal-artin-local-rings-direct-sums-o" rel="nofollow">Keenan Kidwell's question</a> 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.</p>