Using derived categories, we develop an alternative approach to defining Koszulness for positively graded algebras where the degree zero part is not necessarily semisimple.

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.

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...

The purpose of this note is to show how calculi on unital associative algebra with universal right bimodule generalize previously studied constructions by Pusz and Woronowicz [1989] and by Wess and Zumino [1990] and that in this language results are in a natural context, are easier to describe and handle. As a by-product we obtain intrinsic, coordinate-free and basis-independent generalization of the first order noncommutative differential calculi with partial derivatives.

We identify some situations where mappings related to left centralizers, derivations and generalized $(\alpha ,\beta )$-derivations are free actions on semiprime rings. We show that for a left centralizer, or a derivation $T$, of a semiprime ring $R$ the mapping $\psi R\to R$ defined by $\psi \left(x\right)=T\left(x\right)x-xT\left(x\right)$ for all $x\in R$ is a free action. We also show that for a generalized $(\alpha ,\beta )$-derivation $F$ of a semiprime ring $R,$ with associated $(\alpha ,\beta )$-derivation $d,$ a dependent element $a$ of $F$ is also a dependent element of $\alpha +d.$ Furthermore, we prove that for a centralizer $f$ and...

First, we provide an introduction to the theory and algorithms for noncommutative Gröbner bases for ideals in free associative algebras. Second, we explain how to construct universal associative envelopes for nonassociative structures defined by multilinear operations. Third, we extend the work of Elgendy (2012) for nonassociative structures on the 2-dimensional simple associative triple system to the 4- and 6-dimensional systems.