  Let A be a Noetherian local ring with the maximal ideal m and d=dimA. Let Q
be a parameter ideal in A. Let I=Q:m. The problem of when the equality I^2=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, 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^2=QI
holds true for all parameter ideals Q. The result will supply theorems of K.
Yamagishi, S. Goto and K. Nishida with ample examples of ideals I, for which
the Rees algebras R(I), the associated graded rings G(I), and the fiber cones
F(I) are all Buchsbaum rings with certain specific graded local cohomology
modules. Two examples are explored. One is to show that I^2=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^2=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.
