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I'm writing a paper on Emmy Noether for my introductory Abstract Algebra class, and I'm looking for examples of noncommutative Noetherian rings in which the Lasker-Noether theorem fails to hold.

According to the wikipedia article, Noether herself presented a counterexample, although it does not link to any information regarding an example. I can't read German, but maybe it's listed in her 1921 article somewhere?

I'm really just looking one or two simple examples, even if they weren't proposed by Noether herself.

All help is greatly appreciated. Thank you.

Here is the wikipedia article: http://en.wikipedia.org/wiki/Lasker%E2%80%93Noether_theorem

Here is her original paper: http://www.springerlink.com/content/m3457w8h62475473/

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In wiki article there is a direct link to the pdf file of Noether's paper. Did you find out are there any translation of this paper? I searched on the Internet, the result is nothing, but I think such translation should exist. In fact, there is a translation of another paper of Noether, say "Invariant Variation Problems". –  Knumber10 Dec 3 '11 at 9:02
I read the wiki article about Noether but I could not find "Noether herself presented a counterexample, although it does not link to any information regarding an example". Could you please send me this article or cite it here ? Thanks alot. –  Knumber10 Dec 4 '11 at 3:41
Sorry about the delayed response. Tis finals season. The wiki article says " The decomposition does not hold in general for non-commutative Noetherian rings. Noether gave an example of a non-commutative Noetherian ring with a right ideal that is not an intersection of primary ideals." However, even given that clue with the ideals, coming up with an example is a little beyond me. –  JoeDub Dec 6 '11 at 16:36
@Joe: I've merged your accounts. Please register to avoid having to create multiple accounts in the future. –  Qiaochu Yuan Dec 6 '11 at 16:48

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