Commutative Non-Noetherian Rings with the Diamond Property

A ring R is said to have property (◇) if the injective hull of every simple R-module is locally Artinian. By landmark results of Matlis and Vamos, every commutative Noetherian ring has (◇). We give a systematic study of commutative rings with (◇), We give several general characterizations in terms of co-finite topologies on R and completions of R. We show that they have many properties of Noetherian rings, such as Krull intersection property, and recover several classical results of commutative Noetherian algebra, including some of Matlis and Vamos. Moreover, we show that a complete rings has (◇) if and only if it is Noetherian. We also give a few results relating the (◇) property of a local ring with that of its associated graded rings, and construct a series of examples.