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A Laskerian ring is a ring in which every ideal has a primary decomposition. The Lasker-Noether theorem states that every commutative Noetherian ring is a Laskerian ring (as an easy consequence of the ascending chain condition).

And I've found the statement that there are non-Noetherian Laskerian rings, but I can't find an example. Any ideas?

Edit. As the tag already suggested, I'm particularly interested in a commutative Laskerian non-Noetherian ring, but noncommutative examples are also welcome. It never hurts to know more counterexamples.

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3 Answers

up vote 11 down vote accepted

I don't know of any easier examples, but here is one. Let $k$ be a field and $R$ be the set equivalence classes of elements of the form $\frac{f(x,y)}{g(x,y)}$ where $x,y$ are indeterminates over $k$, $f,g\in k[x,y]$, $x$ does not divide $g$ (in $k[x,y]$) and $\frac{f(0,y)}{g(0,y)}\in k$. Make $R$ into a ring by the usual addition and multiplication of rational functions. Then, it can be shown that $R$ is a commutative ring that is not noetherian and every ideal of $R$ has a primary decomposition.

This is an example from Gilmer's paper linked here.

He also gives a characterization of rings in which every ideal has a unique primary decomposition.

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See I. Armeanu, On a class of Laskerian rings, Revue Roum. Math. Pures et Appl. XXII, 8, 1033–1036, Bucharest, 1977.

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Many interesting examples are in Barucci and Fontana: When are $\rm\ D + \frak M\ $ rings Laskerian? They construct (non-Noetherian) Laskerian or strongly Laskerian rings and domains either integrally closed or not, of any dimension. They employ a frequently used powerful tool for constructing counterexamples - the $\rm\ D + \frak M\ $ construction. For more on this construction see Anderson: Star operations and the $\rm\ D + \frak M\ $ construction and see this very informative survey on more general constructions of Zafrullah: Various facets of rings between $\rm\:D[X]\:$ and $\rm\:K[X]\:$.

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@Artur: I have converted your answer to a comment; answers should be reserved for posts that answer the question. But because you do not have 50 reputation points yet, you can only comment on your own questions and answers. So, you didn't do anything wrong; the "add comment" button will only appear for you once you gain 50 points. By the way, here is an explanation of reputation points. –  Zev Chonoles Nov 9 '11 at 19:00
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