On ${\tau }_q$-projectivity and ${\tau }_q$-simplicity

Abstract

In this paper, we first introduce and study the notion of ${\tau }_q$-projective modules via strongly Lucas modules, and then investigate the ${\tau }_q$-global dimension ${\tau }_q$-\gld$(R)$ of a ring $R$. We obtain that if $R$ is a ${\tau }_q$-Noetherian ring, then ${\tau }_q$-\gld$(R)={\tau }_q$-\gld$(R[x])=$\gld$(\mathrm{T}(R[x]))$. Finally, we study the rings over which all modules are ${\tau }_q$-projective (i.e., ${\tau }_q$-semisimple rings). In particular, we show that a ring $R$ is a ${\tau }_q$-semisimple ring if and only if $\mathrm{T}(R[x])$ (or $\mathrm{T}(R)$, or ${\mathcal{Q}}_0(R)$) is a semisimple ring, if and only if $R$ is a reduced ring with $\mathrm{Min}(R)$ finite, if and only if every reg-injective (or semireg-injective, or Lucas, or strongly Lucas) module is injective.