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.
Best Answer
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.