Elementary proof wanted: every local principal ideal ring is a quotient of a PID

2010-05-23T13:45:13Z

I am looking for a more elementary proof of the following result:

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 domain (PID).

Hungerford's article is available free online at:

http://projecteuclid.org/euclid.pjm/1102986148

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).

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).

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?

Answer by NN for Elementary proof wanted: every local principal ideal ring is a quotient of a PID

2012-06-18T05:43:12Z

Theorem 5.2 in http://www.emis.de/journals/BAG/vol.46/no.1/b46h1her.pdf 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)

Elementary proof wanted: every local principal ideal ring is a quotient of a PID - MathOverflow

Elementary proof wanted: every local principal ideal ring is a quotient of a PID

Answer by NN for Elementary proof wanted: every local principal ideal ring is a quotient of a PID