On ${\tau }_q$-flatness and ${\tau }_q$-coherence

Abstract

In this paper, we introduce and study the notions of ${\tau }_q$-flat modules and ${\tau }_q$-coheret rings. First, by investigating the Nagata rings of ${\tau }_q$-torsion theory, we show that the small finitistic dimensions of T$(R[x])$ are all equal to $0$ for any ring $R$. Then, we introduce the notion of ${\tau }_q$-VN regular rings (i.e. over which all modules are ${\tau }_q$-flat), and show that a ring $R$ is a ${\tau }_q$-VN regular ring if and only if T$(R[x])$ is a von Neumann regular ring. Finally, we obtain the Chase theorem for ${\tau }_q$-coheret rings: a ring $R$ is ${\tau }_q$-coherent if and only if any direct product of $R$ is ${\tau }_q$-flat if and only if any direct product of flat $R$-modules is ${\tau }_q$-flat. Some examples are provided to compare with the known conceptions.