Quasi-Hilbert rings and Ratliff-Rush filtrations

Let $A$ be a non Gorenstein Cohen Macaulay ring of dimension $d\geq 1$, $I$ an ideal of $A$, and suppose $\omega_A$ is a canonical $A$-module. Set $$r(I,\omega_A) = \bigcup_{n \geq 0} (I^{n+1} \omega_A : I^{n} \omega_A) \subseteq A .$$ We show that the ideal $r(I,-)$ is $\omega_A$ invariant. Motivated by this property, we introduce a new class of rings, which we call quasi Hilbert rings. We provide several examples of quasi Hilbert rings and discuss a number of their applications. Let $A$ be a local ring with maximal ideal $\mathfrak{m}$. We prove that $A$ is quasi Hilbert iff $\widehat{A}$ is quasi Hilbert, where $\widehat{A}$ is the completion of $A$ w.r.t. $\mathfrak{m}.$ If $d\geq 2$ and $x\in \mathfrak{m}\setminus \mathfrak{m}^2$ is an $A\bigoplus \omega_A$ superficial element, we prove that if $A$ is quasi Hilbert, then so is $A/(x)$. Writing $\widetilde{I}$ for the Ratliff Rush closure of an ideal $I$, we also provide sufficient conditions ensuring the vanishing of $r(I^n,\omega_A)/\widetilde{I^n}$ for all $n\geq 1.$