The Equality I = Qi in Buchsbaum Rings

Let A be a Noetherian local ring with the maximal ideal m and d = dim A. Let Q be a parameter ideal in A. Let I = Q : m. The problem of when the equality I = QI holds true is explored. When A is a Cohen-Macaulay ring, this problem was completely solved by A. Corso, C. Huneke, C. Polini, and W. Vasconcelos [CHV, CP, CPV], while nothing is known when A is not a Cohen-Macaulay ring. The present purpose is to show that within a huge class of Buchsbaum local rings A the equality I = QI holds true for all parameter ideals Q. The result will supply [Y1, Y2] and [GN] with ample examples of ideals I, for which the Rees algebras R(I) = ⊕ n≥0 I n, the associated graded rings G(I) = R(I)/IR(I), and the fiber cones F(I) = R(I)/mR(I) are all Buchsbaum rings with certain specific graded local cohomology modules. Two examples are explored. One is to show that I = QI may hold true for all parameter ideals Q in A, even though A is not a generalized Cohen-Macaulay ring, and the other one is to show that the equality I = QI may fail to hold for some parameter ideal Q in A, even though A is a Buchsbaum local ring with multiplicity at least three.