2023/02/09 by Zhang, Xiaolei
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2302.04560
In this paper, we first introduce and study the notion of τq-projective modules via strongly Lucas modules, and then investigate the τq-global dimension τq-\gld(R) of a ring R. We obtain that if R is a τq-Noetherian ring, then τq-\gld(R)=τq-\gld(R[x])=\gld(\rm T(R[x])). Finally, we study the rings over which all modules are τq-projective (i.e., τq-semisimple rings). In particular, we show that a ring R is a τq-semisimple ring if and only if \rm T(R[x]) (or \rm T(R), or Q0(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.