In Section 1, we will review closure operations on the set of ideals, the set of fractional ideals and the set of R-submodules of the total ring of fractions of a commutative ring.
Recall that a fractional ideal of R is an R-submodule A of Q satisfying the property that there exists a regular element x R
We denote the R-submodules of Q by (R), the fractional ideals of R by (R) and the ﬁnitely generated fractional ideals by f (R).
A closure operation on the set of ideals of a commutative ring R (respectively the set of fractional ideals of R or the R-submodules of Q) is a function c : I(R) → I(R) (respectively c : F(R) → F(R)
A closure operation on the R-submodules of Q is termed divisible if for all regular elements u of Q and all R-submodules A of Q, (uA)c = uAc.
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