If $\tau$ is a hereditary torsion theory on ${\mathrm{𝐌𝐨𝐝}}_R$ and $Q_{\tau }:{\mathrm{𝐌𝐨𝐝}}_R\to {\mathrm{𝐌𝐨𝐝}}_R$ is the localization functor, then we show that every $f$-derivation $d:M\to N$ has a unique extension to an $f_{\tau }$-derivation $d_{\tau }:Q_{\tau }\left(M\right)\to Q_{\tau }\left(N\right)$ when $\tau$ is a differential torsion theory on ${\mathrm{𝐌𝐨𝐝}}_R$. Dually, it is shown that if $\tau$ is cohereditary and $C_{\tau }:{\mathrm{𝐌𝐨𝐝}}_R\to {\mathrm{𝐌𝐨𝐝}}_R$ is the colocalization functor, then every $f$-derivation $d:M\to N$ can be lifted uniquely to an $f_{\tau }$-derivation $d_{\tau }:C_{\tau }\left(M\right)\to C_{\tau }\left(N\right)$.

This is a summary of some of the main results in the monograph Faithfully Ordered Rings (Mem. Amer. Math. Soc. 2015), presented by the first author at the ALANT conference, Będlewo, Poland, June 8-13, 2014. The notions involved and the results are stated in detail, the techniques employed briefly outlined, but proofs are omitted. We focus on those aspects of the cited monograph concerning (diagonal) quadratic forms over preordered rings.

Let $R$ be a ring. A left $R$-module $M$ is called an FC-module if $M^{+}={Hom}_{\mathbb{Z}}(M,\mathbb{Q}/\mathbb{Z})$ is a flat right $R$-module. In this paper, some homological properties of FC-modules are given. Let $n$ be a nonnegative integer and ${\mathrm{ℱ𝒞}}_n$ the class of all left $R$-modules $M$ such that the flat dimension of $M^{+}$ is less than or equal to $n$. It is shown that $({}^{\perp }\left({\mathrm{ℱ𝒞}}_n^{\perp }\right),{\mathrm{ℱ𝒞}}_n^{\perp })$ is a complete cotorsion pair and if $R$ is a ring such that $fd\left({\left({}_{R}R\right)}^{+}\right)\le n$ and ${\mathrm{ℱ𝒞}}_n$ is closed under direct sums, then $({\mathrm{ℱ𝒞}}_n,{\mathrm{ℱ𝒞}}_n^{\perp })$ is a perfect cotorsion pair. In particular, some known results are obtained as corollaries....

We provide a technique to find a cluster-tilting object having a given cluster-tilted algebra as endomorphism ring in the finite type case.

We give an automata-theoretic description of the algebraic closure of the rational function field ${𝔽}_q\left(t\right)$ over a finite field ${𝔽}_q$, generalizing a result of Christol. The description occurs within the Hahn-Mal’cev-Neumann field of “generalized power series” over ${𝔽}_q$. In passing, we obtain a characterization of well-ordered sets of rational numbers whose base $p$ expansions are generated by a finite automaton, and exhibit some techniques for computing in the algebraic closure; these include an adaptation to positive...