A Lasker ring is a ring in which every ideal has a primary decomposition. The Lasker-Noether theorem states that every commutative Noetherian ring is a Lasker ring (as an easy consequence of the ascending chain condition). And I've found the statement that there are non-Noetherian Lasker rings, but I can't find an example.
edit: As the tag already suggested, I'm particularly interested in a commutative non-Noetherian Lasker ring, but noncommutative examples are also welcome. It never hurts to know more counterexamples.