# A Laskerian non-Noetherian ring

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.

## 3 Answers

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.

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 $\rm\:K[X]\:$.

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