  This is the first in a series of papers highlighting the applications of
reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ an
ideal of $R$. We show that $I$-reduced $R$-modules and $I$-coreduced
$R$-modules provide a setting in which the Matlis-Greenless-May (MGM)
Equivalence and the Greenless-May (GM) Duality hold. These two notions have
been hitherto only known to exist in the derived category setting. We realise
the $I$-torsion and the $I$-adic completion functors as representable functors
and under suitable conditions compute natural transformations between them and
other functors.
