# ${\rm cl}$-prereductions, ${\rm i}$-postexpansions, and related structures

## TL;DR

This paper develops a dual theory of prereductions and postexpansions for modules over Noetherian rings, establishing duality relations and exploring their properties and classifications within the context of closure and interior operations.

## Contribution

It introduces the concepts of ${
m cl}$-prereductions and ${
m i}$-postexpansions, establishing their duality and analyzing their structural properties and classifications.

## Key findings

- Duality between ${
m cl}$-prereductions and ${
m i}$-postexpansions established.
- Comparison of ${
m cl}$-core and ${
m i}$-hull with prereductions and postexpansions.
- Classification of prereductions of ${
m cl}$-closed ideals in Noetherian rings.

## Abstract

Expanding on the work of Kemp, Ratliff and Shah, for any closure ${\rm cl}$ defined on a class of modules over a Noetherian ring, we develop the theory of ${\rm cl}$-prereductions of submodules. For any interior ${\rm i}$ on a class of $R$-modules, we also develop the theory of {\rm i}-postexpansions. Using the duality of Epstein, R.G. and Vassilev, we show that if ${\rm i}$ is the interior dual to ${\rm cl}$, then these notions are in fact dual to each other. We consider the ${\rm cl}$-precore (${\rm i}$-postcore), the intersection of all ${\rm cl}$-prereductions ${\rm i}$-postexpansions) of a submodule and the ${\rm cl}$-prehull (${\rm i}$-posthull), the sum of all ${\rm cl}$-prereductions (${\rm i}$-postexpansions) of a submodule and give comparisons with the ${\rm cl}$-core (${\rm i}$-hull). We further give a classification of ${\rm cl}$-prereductions of ${\rm cl}$-closed ideals of a Noetherian ring where ${\rm cl}$ is a closure with a special part.