On τq-projectivity and τq-simplicity

In this paper, we first introduce and study the notion of [Formula: see text]-projective modules via strongly Lucas modules, and then investigate the [Formula: see text]-global dimension [Formula: see text]-gl.dim[Formula: see text] of a ring [Formula: see text]. We obtain that if [Formula: see text] is a [Formula: see text]-Noetherian ring, then [Formula: see text]-gl.dim[Formula: see text]-gl.dim[Formula: see text]gl.dim[Formula: see text]. Finally, we study the rings over which all modules are [Formula: see text]-projective (i.e., [Formula: see text]-semisimple rings). In particular, we show that a ring [Formula: see text] is a [Formula: see text]-semisimple ring if and only if [Formula: see text] (or [Formula: see text], or [Formula: see text]) is a semisimple ring, if and only if [Formula: see text] is a reduced ring with [Formula: see text] finite, if and only if every reg-injective (or semireg-injective, or Lucas, or strongly Lucas) module is injective.