Integral closure, basically full closure, and duals of nonresidual closure operations

How to Extend Closure and Interior Operations to More Modules

“…Suppose that p and q are pair operations on a class of pairs P. We say that p ≤ q if for all (L,M ) ∈ P, p(L,M ) ⊆ q(L,M ). As we have pointed out both in this and our previous paper [9], pair operations are a generalization of submodule selectors. In [8,Prop.…”

“…In [9], we defined a new version of duality that works on any pair operation, a notion that encompasses both closure and interior operations. In particular, we used this duality to find the duals of non-residual closures, which are relative interior operations.…”

“…In this paper, we generalize the three methods of extending a closure operation to the context of pair operations, as well as discussing how to convert a closure operation to a hereditary or residual one. In §2, we recall the relevant properties of pair operations that we introduced in [9] along with a number of versions of integral closure and tight closure. In §3, we discuss some new properties for pair operations.…”

“…In §6, we show that when the ring (R,m) is a complete local ring, then the cohereditary and hereditary versions of pair operations that we constructed in § §4 and 5 are dual to each other. In §7, we continue the study of J -basically full closures and J -basically empty interiors we began in [9], by discussing the residual version of the J -basically full closure (which is a hereditary closure) and the absolute version of the J -basically empty interior (which is a cohereditary interior). In §8, we introduce pair operations defined by pre-enveloping classes.…”

“…In §9, we analyze known closures such as integral closure and tight closure through the lens of pre-enveloping classes. In [9], for any ideal J, we discussed the Jbf-closure and introduced the Jbe-interior. In Propositions 7.3 and 7.12, we show that Jbf is hereditary and Jbe is cohereditary.…”