On $\tau_q$-projectivity and $\tau_q$-simplicity

TL;DR

This paper introduces and explores $ au_q$-projective modules and $ au_q$-semisimple rings, establishing their properties, global dimensions, and characterizations in relation to ring structures and module injectivity.

Contribution

It defines $ au_q$-projective modules via strongly Lucas modules, investigates their global dimension, and characterizes $ au_q$-semisimple rings through ring and module properties.

Findings

$ au_q$-global dimension equals classical global dimension for $ au_q$-Noetherian rings.

A ring is $ au_q$-semisimple iff certain associated rings are semisimple.

All modules are $ au_q$-projective iff the ring is reduced with finitely many minimal primes.

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$({\rm T}(R[x]))$. 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 ${\rm T}(R[x])$ (or ${\rm T}(R)$, or $\mathcal{Q}_0(R)$) is a semisimple ring, if and only if $R$ is a reduced ring with ${\rm Min}(R)$ finite, if and only if every reg-injective (or semireg-injective, or Lucas, or strongly Lucas) module is injective.