On τq-flatness and τq-coherence

In this paper, we first introduce and study the notions of [Formula: see text]-flat modules. Then study the rings over which all modules are [Formula: see text]-flat (which is called [Formula: see text]-VN regular rings). In particular, we show that a ring [Formula: see text] is a [Formula: see text]-VN regular ring if and only if T[Formula: see text] is a von Neumann regular ring, if and only if [Formula: see text] is a reduced ring with Min[Formula: see text] compact. Finally, we obtain the Chase theorem for [Formula: see text]-coheret rings: a ring [Formula: see text] is [Formula: see text]-coherent if and only if any direct product of [Formula: see text] is [Formula: see text]-flat if and only if any direct product of flat [Formula: see text]-modules is [Formula: see text]-flat. Some examples are provided to compare with the known conceptions.