We study modules whose endomorphism rings possess various forms of von Neumann regularity. We characterize these “regularity” properties for several classes of modules, including completely decomposable modules and finitely presented modules over commutative rings. We generalize many of the classic results about abelian groups with regular endomorphism rings to modules over one-dimensional commutative rings with Noetherian spectrum. To facilitate this study, we define a module M over a commutative ring R to be weakly endoregular if xM and $(0{:}_M(x))$ are direct summands of M for each $x\in R$. We give several characterizations of weak endoregularity for various classes of modules.

我们研究其内同态环具有各种形式的冯·诺依曼规则性的模块。我们对几种类型的模块（包括完全可分解的模块和交换环上的有限表示的模块）的这些“规则性”特性进行了表征。我们将具有规则内同态环的阿贝尔群的许多经典结果推广到具有Noetherian谱的一维交换环上的模块。为了促进这项研究，我们将交换环R上的模M定义为弱内规，如果xM和$（0{：}_{\mathrm{中号}}（X））$是每个M的直接求和$X\in \mathrm{[R}$。我们给出了各种类别模块的弱内规整度的几个特征。