On the Integral Closures of Ideals

The Tight Integral Closure of a Set of Ideals Relative to Modules

“…Definition 3.4. (See [1]). A subset T of Ass(R) has reduced property if for every P ∈ T , there exists an element x ∈ R such that P = Ann(x) and x 2 = 0.…”

“…From [1], [3], and [8] we know that I * (M ) = {x ∈ R : x is integrally dependent on I relative to M } is an ideal of R.…”

“…Let M be an R−module. The dual notion of integral closure of ideals relative to modules was introduced in [1].…”