Nakayama closures, interior operations, and core-hull duality -- with applications to tight closure theory (Q2084536)

The results in the paper mostly concern modules over commutative rings, but some hold for non-commutative rings also. Rings with special properties are utilized in different contexts. The springboard seems to be a 2021 paper by the first two authors, where a number of notions had been defined ([\textit{N. Epstein} and \textit{R. R. G.}, Rocky Mt. J. Math. 51, No. 3, 823--853 (2021; Zbl 1477.13047)]). All the terminology used may be found in the paper under review. The authors prove that the interiors of ideals may be computed in Noetherian local rings (that are approximately Gorenstein) using the dual closure and colons; they show that, in case of Gorenstein rings, if parameter ideals are cl-closed, then all ideals are cl-closed. It is shown that a number of module and algebra closures are Nakayama. Nakayama closures are shown to be dual to Nakayama interiors. A number of constructs are defined and examined, such as i-expansions, finite co-generation, minimal co-generating sets, maximal expansions and the core-hull duality. Related closures and interiors are compared, the co-spread is defined as a dual spread and it is shown that liftable integral spreads exist. For a taste of the results we quote the following Theorem 8.1: Assume that (\(R, \mathfrak{m}\)) is a complete reduced \(F\)-finite local ring and \(I\) is an ideal of \(R\). Assume also that either of the following two conditions hold: \begin{itemize} \item[1.] There is a positively graded \(\mathbb{N}\)-graded algebra \(A\) over a field \(K\), with graded maximal ideal \(\mathfrak{n}\), and a homogeneous ideal \(J\) of \(A\), such that \(R=\hat A_\mathfrak{n}\) and \(I=JR\), or \item[2.] \(R\) is an isolated singularity. \end{itemize} Then, the tight interior of \(I\) and the Artinistic version of the tight interior of \(I\) coincide. This (and other) result is then used to compute some examples of tight and Frobenius interiors and hulls.