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

Abstract

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

26 pages, 2 figures