Some algebraic and homological properties of a family of quotients of the Rees algebra

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a proper ideal of [Formula: see text]. In this paper, we study some algebraic and homological properties of a family of rings [Formula: see text], with [Formula: see text], that are obtained as quotients of the Rees algebra associated with the ring [Formula: see text] and the ideal [Formula: see text]. Specially, we study when [Formula: see text] is a von Neumann regular ring, a semisimple ring and a Gaussian ring. Also, we study the classical global and weak global dimensions of [Formula: see text]. Finally, we investigate some homological properties of [Formula: see text]-modules and we show that [Formula: see text] and [Formula: see text] are Gorenstein projective [Formula: see text]-modules, provided some special conditions.