Asymptotic behavior of parameter ideals in generalized Cohen–Macaulay modules

The purpose of this paper is to give affirmative answers to two open questions as follows. Let be a generalized Cohen–Macaulay Noetherian local ring. Both questions, the first question was raised by M. Rogers [M. Rogers, The index of reducibility for parameter ideals in low dimension, J. Algebra 278 (2004) 571–584] and the second one is due to S. Goto and H. Sakurai [S. Goto, H. Sakurai, The equality I2=QI in Buchsbaum rings, Rend. Sem. Mat. Univ. Padova 110 (2003) 25–56], ask whether for every parameter ideal contained in a high enough power of the maximal ideal the following statements are true: (1) The index of reducibility is independent of the choice of ; and (2) , where .