elementary classification of artinian rings - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T04:03:03Z http://mathoverflow.net/feeds/question/10941 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10941/elementary-classification-of-artinian-rings elementary classification of artinian rings Martin Brandenburg 2010-01-06T18:07:09Z 2010-06-25T20:34:12Z <p>this may be too elementary for mathoverflow, but I'll give it a try.</p> <p>rings are commutative here. it is well-known that every \$0\$-dimensional noetherian ring is artinian. the standard proof uses a filtration argument; then it's left to show that every finitely generated vector space is artinian (dimension!) and that extensions of artinian by artinian modules are artinian (tage the images and the preimages of the chain, finally both are stable). by a sheaf argument, it's easy to reduce to: every noetherian ring with exactly one prime ideal is artinian.</p> <p><b>is there a proof which is somehow more direct?</b> perhaps a clever manipulation of chains of ideals? I don't expect it, but it would be great for the students in my tutorial, which had to solve this as an exercise without knowing anything about artinian or noetherian rings going beyond the definitions.</p> http://mathoverflow.net/questions/10941/elementary-classification-of-artinian-rings/10943#10943 Answer by Pete L. Clark for elementary classification of artinian rings Pete L. Clark 2010-01-06T18:19:29Z 2010-01-06T18:19:29Z <p>My take on the standard proof can be found on pp. 62-63 of my notes on commutative rings:</p> <p><a href="http://math.uga.edu/~pete/integral.pdf" rel="nofollow">http://math.uga.edu/~pete/integral.pdf</a></p> <p>(The bit about ACC/DCC being preserved by extensions occurs on p. 57 and should probably be explicitly mentioned in the proof on p. 63.) </p> <p>Altogether this takes about 1-1.5 pages. I have never seen anything substantially shorter or more direct. </p> <p>By the way, that's a tough problem to ask a student to solve on his/her own!</p> http://mathoverflow.net/questions/10941/elementary-classification-of-artinian-rings/29542#29542 Answer by Greg Marks for elementary classification of artinian rings Greg Marks 2010-06-25T20:34:12Z 2010-06-25T20:34:12Z <p>Sorry if this is merely a reformulation of what has already been said (and doubtless it is a "standard proof"), but perhaps a suggestive hint for students would be to show that if an ideal \$P\$ of a commutative noetherian ring \$R\$ is maximal for the property that \$R/P\$ is non-artinian, then \$P \subset R\$ must be prime.&nbsp; A sort of philosophical underpinning for this hint is offered in a pretty paper by T. Y. Lam and M. L. Reyes, "A prime ideal principle in commutative algebra," <em>J. Algebra</em> <strong>319</strong> (2008), no. 7, 3006-3027.</p>