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.

On $\tau_q$-flatness and $\tau_q$-coherence

Let $R$ be a commutative ring with identity. The small finitistic dimension $\fPD(R)$ of $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we characterize a ring $R$ with $\fPD(R)\leq n$ using finitely generated semi-regular ideals, tilting modules, cotilting modules of cofinite type or vaguely associated prime ideals. As an application, we obtain that if $R$ is a Noetherian ring, then $\fPD(R)= \sup\{\grade(\m,R)|\m\in \Max(R)\}$ where $\grade(\m,R)$ is the grade of $\m$ on $R$ . We also show that a ring $R$ satisfies $\fPD(R)\leq 1$ if and only if $R$ is a $\DW$ ring. As applications, we show that the small finitistic dimensions of strong \Prufer\ rings and $\LPVD$s are at most one. Moreover, for any given $n\in \mathbb{N}$, we obtain examples of total rings of quotients $R$ with $\fPD(R)=n$.

The small finitistic dimensions over commutative rings

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The small finitistic dimensions over commutative rings