Homological properties of a family of quotients of the Rees algebra

Let $R$ be a commutative ring, $a,b$ be elements of $R$, and let $I\neq0$ be a proper ideal of $R$. We investigate some homological properties of the family of quotients of the Rees algebra $R(I)_{a,b}$, which was introduced by Barucci et al. In particular, we study the transferring of being partial $n$-tilting, Gorenstein projective, Gorenstein injective and Gorenstein flat between $R$ and $R(I)_{a,b}$.