  Let (R,m) be a noetherian local ring and let $\mathcal{C}$ be the class of
all R-modules M which possess a reflexive submodule U such that M/U is finitely
generated. For every R-module $M\in \mathcal{C}$ the canonical embedding
$\varphi: M\to M^{oo}$ is pure-essential. We investigate in the first section
under which conditions the reverse is true, for example if R is a discrete
valuation ring or if R does not have nilpotent elements and M is flat. In
section 2 we determine all reflexive and flat R-modules with the help of a
certain analogy between the localization $R_q$ and the injective hull of R/q.
In section 3 we show: If the property 'pure-essential' is transitive for a
domain R, then it follows that $dim(R)\leq 1$.
