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.

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be a $u$-$S$-Artinian module if there is $s\in S$ such that any descending chain of submodules of $M$ is $S$-stationary with respect to $s$. The notion of $u$-$S$-Artinian modules is characterized in terms of ($S$-MIN)-conditions and $u$-$S$-cofinite properties. A ring $R$ is said to be $u$-$S$-Artinian if $R$ itself is a $u$-$S$-Artinian module. It is then shown that any $u$-$S$-semisimple ring is $u$-$S$-Artinian. It is proved that a ring $R$ is $u$-$S$-Artinian if and only if $R$ is $u$-$S$-Noetherian, the $u$-$S$-Jacobson radical $\operatorname{Jac}_S(R)$ of $R$ is $S$-nilpotent and $R/\operatorname{Jac}_S(R)$ is a $u$-$S/\operatorname{Jac}_S(R)$-semisimple ring. Besides, some examples are given to distinguish Artinian rings, $u$-$S$-Artinian rings, and $S$-Artinian rings.

We establish several sufficient conditions under which a locally-integrable function $f:\mathbb{R}^n \to \mathbb{R}$ represents a positive-definite distribution. In particular, we consider functions of the form $f(\|x\|)$, where $\|\cdot\|$ is a fixed norm in $\mathbb{R}^n$.

We investigate several properties of Bott–Duffin $(e, f)$ inverses in $e$-symmetric rings. In particular, we present new characterizations of Bott–Duffin $(e, f)$ inverses in terms of units in such rings. We also study the EP property of an element $a$ in an $ab$-symmetric ring, where $b$ is a generalized inverse of $a$. Finally, we consider the Bott–Duffin decomposition associated with idempotents.

The subdivision of an ordinary graph is obtained by inserting a vertex into every edge, while its $R$-graph is obtained by adding a new vertex to every edge and joining it to both ends. We study the corresponding constructions for signed graphs, prove that both are stable under switching, resolve their balance, and determine the spectrum (of the adjacency matrix) of the signed $R$-graph in the regular case. We also introduce two corona-like products based on signed subdivisions and four based on signed $R$-graphs. We determine their characteristic polynomials and spectra, together with their Laplacian characteristic polynomials and spectra, either in general or under appropriate regularity assumptions. Finally, for the generalized subdivision introduced in [Ars Math. Contemp. 23 (2023), 3–9], we express its spectrum in terms of the Laplacian spectrum of the original signed graph, thereby answering a problem posed there. These results provide constructions of cospectral and Laplacian cospectral signed graphs.