Jayaram and Tekir defined an R-module M, R is a commutative ring, to be “von Neumann regular” if for each there exists an such that Previously, Fieldhouse called M “regular” if every submodule is pure and Ramamurthi and Rangaswamy called M “strongly regular” if every finitely generated submodule is a direct summand. We call these three notions JT-regular, F-regular, and strongly F-regular, respectively. We define M to be almost locally simple if for each maximal ideal of R, is either a trivial or simple -module and weakly JT-regular if for each We show that JT-regular almost locally simple strongly F-regular F-regular weakly JT-regular and investigate when these implications can be reversed. We provide some new characterizations of these properties and investigate each property in the context where M is finitely generated or R is Dedekind or more generally J-Noetherian.