This paper is a survey about ring theory properties of some quotients of Rees algebras \(R_f(\mathfrak i)\) , where R is a unitary commutative ring, \(f \in R[T]\) and \(\mathfrak i\) is an ideal of R. More precisely \(R_f(\mathfrak i) :=R[\mathfrak i T]/(fR[T]\cap R[\mathfrak i T]).\) These rings can be studied using pullback constructions (this is very useful to describe the prime spectrum and to investigate some questions related to the structure of the ideals). We start by discussing the case when the polynomial f has degree 2 and in the last section we generalize to constructions obtained with polynomials of any degree.