Abstract
Let
R
be a commutative Noetherian ring, and let
\mathbf{x}= x_1, \ldots, x_d
be a regular
R
-sequence contained in the Jacobson radical of
R
. An ideal
I
of
R
is said to be a monomial ideal with respect to
\mathbf{x}
if it is generated by a set of monomials
x_1^{e_1}\ldots x_d^{e_d}
. The monomial closure of
I
, denoted by
\widetilde{I}
, is defined to be the ideal generated by the set of all monomials
m
such that
m^n\in I^n
for some
n\in \mathbb{N}
. It is shown that the sequences
\mathrm{Ass}_RR/\widetilde{I^n}
and
\mathrm{Ass}_R\widetilde{I^n}/I^n
,
n=1,2, \ldots,
of associated prime ideals are increasing and ultimately constant for large
n
. In addition, some results about the monomial ideals and their integral closures are included.