论$τ_q$-投射性与$τ_q$-单性
On $τ_q$-projectivity and $τ_q$-simplicity
AI总结：
本文引入$τ_q$-投射模概念，研究$τ_q$-整体维数在$τ_q$-Noetherian环下的等式关系，并刻画$τ_q$-半单环的多种等价条件。
摘要：
本文首先通过强Lucas模引入并研究了$τ_q$-投射模的概念，随后探究了环$R$的$τ_q$-整体维数$τ_q$-gld$(R)$。我们得到：若$R$是$τ_q$-Noetherian环，则$τ_q$-gld$(R)=τ_q$-gld$(R[x])=$gld$({\rm T}(R[x]))$。最后，我们研究了所有模都是$τ_q$-投射模的环（即$τ_q$-半单环）。特别地，我们证明了环$R$是$τ_q$-半单环当且仅当${\rm T}(R[x])$（或${\rm T}(R)$，或$\ncal{Q}_0(R)$）是半单环，当且仅当$R$是${\rm Min}(R)$有限的约化环，当且仅当每个reg-内射（或半正则内射，或Lucas，或强Lucas）模都是内射模。
英文摘要：
In this paper, we first introduce and study the notion of $τ_q$-projective modules via strongly Lucas modules, and then investigate the $τ_q$-global dimension $τ_q$-\gld$(R)$ of a ring $R$. We obtain that if $R$ is a $τ_q$-Noetherian ring, then $τ_q$-\gld$(R)=τ_q$-\gld$(R[x])=$\gld$({\rm T}(R[x]))$. Finally, we study the rings over which all modules are $τ_q$-projective (i.e., $τ_q$-semisimple rings). In particular, we show that a ring $R$ is a $τ_q$-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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
