Regular completion of modules

In this paper we introduce the concept of a module regular in the sense of von Neumann. We construct a regular completion of a module $X$ torsion-free relative to a filter $\mathfrak F_R$of dense modules over a commutative semiprimary ring $R$. The paper's main result is a theorem that module $X$ is divisible (is injective relative to filter $\mathfrak F_R$) if and only if it is von Neumann-regular and orthocomplete. We prove that a divisible hull of module $X$ relative to $\mathfrak F_R$ is a composition of two simpler completions: a regular one and an orthocompletion.