Elementary proof wanted: every local principal ideal ring is a quotient of a PID - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:53:58Z http://mathoverflow.net/feeds/question/25663 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25663/elementary-proof-wanted-every-local-principal-ideal-ring-is-a-quotient-of-a-pid Elementary proof wanted: every local principal ideal ring is a quotient of a PID Pete L. Clark 2010-05-23T13:45:13Z 2012-06-18T05:43:12Z <p>I am looking for a more elementary proof of the following result:</p> <p>Theorem (Hungerford, 1968): Let \$R\$ be a principal ideal ring. Then \$R \cong \prod_{i=1}^n R_i\$, where each \$R_i\$ is a homomorphic image of a principal ideal <em>domain</em> (PID).</p> <p>Hungerford's article is available free online at:</p> <p><a href="http://projecteuclid.org/euclid.pjm/1102986148" rel="nofollow">http://projecteuclid.org/euclid.pjm/1102986148</a></p> <p>What do I mean by "more elementary"? Hungerford uses the Cohen structure theory of complete local rings, which I would like to avoid (because I have notes on commutative algebra which do not discuss such things). </p> <p>Note that Hungerford's theorem is a refinement of a previous result of Zariski and Samuel, which asserts that a principal ideal ring is isomorphic to a finite direct product of rings, each of which is either a PID or a "special principal ideal ring", i.e., a local Artinian principal ideal ring. The proof of this result uses primary decomposition, which is acceptable to me (in fact I put a section on primary decomposition into my notes for exactly this application). </p> <p>Given the theorem of Zariski-Samuel, Hungerford's result is plainly equivalent to the fact that every Artinian local principal ideal ring is the quotient of a PID. Now doesn't that sound like you should be able to prove it without invoking the structure theory of complete local rings? </p> http://mathoverflow.net/questions/25663/elementary-proof-wanted-every-local-principal-ideal-ring-is-a-quotient-of-a-pid/99872#99872 Answer by NN for Elementary proof wanted: every local principal ideal ring is a quotient of a PID NN 2012-06-18T05:43:12Z 2012-06-18T05:43:12Z <p>Theorem 5.2 in <a href="http://www.emis.de/journals/BAG/vol.46/no.1/b46h1her.pdf" rel="nofollow">http://www.emis.de/journals/BAG/vol.46/no.1/b46h1her.pdf</a> gives an answer (take the projective limit. The paper has a related one with corrections, but not for the part that is related to your question). This is for a non-commutative case, and the theorem has a non-commutative extension: a PIR is a finite direct product of prime and artinian indecomposable cases, which are matrix rings over CPU rings (Faith, Algebra II should contain all the needed references)</p>