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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 classof $R$-modules, we also develop the theory of {\rm i}-postexpansions. Using theduality of Epstein, R.G. and Vassilev, we show that if ${\rm i}$ is theinterior 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 ofall ${\rm cl}$-prereductions ${\rm i}$-postexpansions) of a submodule and the${\rm cl}$-prehull (${\rm i}$-posthull), the sum of all ${\rmcl}$-prereductions (${\rm i}$-postexpansions) of a submodule and givecomparisons with the ${\rm cl}$-core (${\rm i}$-hull). We further give aclassification of ${\rm cl}$-prereductions of ${\rm cl}$-closed ideals of aNoetherian ring where ${\rm cl}$ is a closure with a special part.

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