The purpose of this paper is to give affirmative answers to two open questions as follows. Let
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be a generalized
Cohenx2013;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
x2013;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
x2013;56], ask whether for every parameter ideal
![]()
contained in a high enough power of the
maximal ideal
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the following statements are true: (1) The index of reducibility
![]()
is independent of the choice of
![]()
; and (2)
![]()
, where
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.