In this paper author investigates the properties of PI-rings having faithful module with Krull dimension relative to a noetherian torsion theory. The main results of this paper: Let $R$ be an associative PI-ring with identity, M be a left faithful $R$-module, $\tau$ — noetherian torsion theory. Let $\tau M=0$ and module $M$ have $\tau$-Krull dimension. If $N$ is a nil ideal then there exists a natural $n$ such that ${N}^{n}M=0$. Let $R$ be an associative PI-ring with identity, $M$ be a left faithful $R$-module, $\tau$ — noetherian torsion theory. Let module $M$ have $\tau$-Krull dimension. If $R$ is $\tau$-torsionfree as left $R$-module, module $M$ and prime radical of $R$ are finitely generated, then $R$ has left $\tau$-Krull dimension and left $\tau$-Krull dimension of $R$ is equal to left $\tau$-Krull dimension of module $M$.