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I am wondering if there are applications of non-noetherian rings (domains). Are there areas where it is applicable ? Now obviously there is an academic interest in it. But I want to know if there are areas where one might find an application, such as in algebraic statistics etc.

Thanks in advance.

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You put "domains" in parentheses... does that mean you only want domains or ...? –  rschwieb Mar 5 '14 at 23:17

2 Answers 2

Often function rings are not noetherian and these sorts of rings come up frequently in applications, though what one counts as an application in mathematics obviously depends on who you are talking to. For instance, the ring of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ is not noetherian.

Even more frequently we see applications of non artinian rings, as the integers even are non artinian, for instance.

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von Neumann regular rings are pretty important in functional analysis, and they are Noetherian iff they are semisimple rings, so the "interesting" ones are the non-Noetherian ones.

I should also say that this class of examples is kind of orthogonal to the other (also very good) example chstan gave about rings of continuous functions. The ring of continuous real functions on a connected compact Hausdorf space with more than one element is never von Neumann regular.

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