Direct complements in a module $M$ are said to be almost (essentially) unique if, whenever $M = A \oplus B = A \oplus C$, $(B+C)/B$ is small in $M/B$ ($B \cap C$ is essential in $B$). The module $M$ is said to be a DCAU-module (DCEU-module) if direct complements of $M$ are almost (essentially) unique. We determine the structure of both DCAU- and DCEU-modules over discrete valuation rings. When $R$ is a non-local Dedekind domain, we describe the structure of the torsion part of a DCAU-$R$-module $M$ (and of a DCEU-$R$-module $N$) which turns out to be a direct summand of $M$ (of $N$). Moreover, we investigate the class of rings $R$ for which every right DCAU-$R$-module is DCEU. A ring of this type will be called a right AE-ring. Analogous to this class of rings, we shed some light on right EA-rings (i.e., rings $R$ for which every right DCEU-$R$-module is DCAU). Among other results, we show that every right AE-ring is right Bass and every commutative EA-ring is perfect. We provide examples to delineate the concepts and results.

We demonstrate that three maximal subgroups of infinite index in the rectangular subgroup $K_{(2,2)}$ of the Thompson group $F$, each containing Jones's 3-colorable subgroup $\mathcal{F}$, can be characterized as stabilizer subgroups. Additionally, we show that the $\vec{F}$-index, an elementary knot invariant introduced thanks to Jones's construction of knots from Thompson groups, may increase by at most 3 after changing the orientation of a knot.