Abstract. Let A be a commutative ring and I an ideal of A with a reduction Q. In this paper we give an upper bound on the reduction number of I with respect to Q, when a suitable family of ideals in A is given. As a corollary it follows that if some ideal J containing I satisfies J 2 = QJ, then I v+2 = QI v+1 , where v denotes the number of generators of J/I as an A-module.

Abstract. Let F = {F n } be a multiplicative filtration of a local ring such that the Rees algebra R(F ) is Noetherian. We recall Burch's inequality for F and give an upper bound of the a-invariant of the associated graded ring a(G(F )) using a reduction system of F . Applying those results, we study the symbolic Rees algebra of certain ideals of dimension 2.