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

An $R$-module $M$ is said to be an extending module if every closed submodule of $M$ is a direct summand. In this paper we introduce and investigate the concept of a type 2 $\tau$-extending module, where $\tau$ is a hereditary torsion theory on $\text{Mod}$-$R$. An $R$-module $M$ is called type 2 $\tau$-extending if every type 2 $\tau$-closed submodule of $M$ is a direct summand of $M$. If ${\tau }_I$ is the torsion theory on $\text{Mod}$-$R$ corresponding to an idempotent ideal $I$ of $R$ and $M$ is a type 2 ${\tau }_I$-extending $R$-module, then the question of whether or not $M/MI$ is...

It is shown that a ring $R$ is a $GM$-ring if and only if there exists a complete orthogonal set $\{e_1,\cdots ,e_n\}$ of idempotents such that all $e_{i}R{e}_i$ are $GM$-rings. We also investigate $GM$-rings for Morita contexts, module extensions and power series rings.

We characterize linear operators that preserve sets of matrix ordered pairs which satisfy extreme properties with respect to maximal column rank inequalities of matrix sums over semirings.

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 K be a field of characteristic p > 0, K* the multiplicative group of K and $G=G_p\times B$ a finite group, where $G_p$ is a p-group and B is a p’-group. Denote by $K^{\lambda }G$ a twisted group algebra of G over K with a 2-cocycle λ ∈ Z²(G,K*). We give necessary and sufficient conditions for G to be of OTP projective K-representation type, in the sense that there exists a cocycle λ ∈ Z²(G,K*) such that every indecomposable $K^{\lambda }G$-module is isomorphic to the outer tensor product V W of an indecomposable $K^{\lambda }{G}_p$-module V and a simple...