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.

如果每个简单R模块的内射壳都是局部Artinian，则称环R具有属性（◇）。根据Matlis和Vamos的划时代结果，每个可交换Noetherian环都有（◇）。我们给交换环与系统的研究（◇），我们给几个一般性的刻画在联合有限拓扑方面[R和完井[R。我们证明它们具有Noetherian环的许多特性，例如Krull交集特性，并恢复了交换Noetherian代数的一些经典结果，包括Matlis和Vamos中的一些。此外，我们证明，当且仅当它是Noetherian时，一个完整的环才具有（◇）。我们还给出了一些有关局部环的（◇）属性与其关联的渐变环的属性的结果，并构建了一系列示例。