  Geoffrey Dietz introduced a set of axioms for a closure operation on a
complete local domain R so that the existence of such a closure operation is
equivalent to the existence of a big Cohen-Macaulay module. These closure
operations are called Dietz closures. In complete rings of characteristic p >
0, tight closure and plus closure satisfy the axioms. In order to study these
closures, we define module closures and discuss their properties. For many of
these properties, there is a smallest closure operation satisfying the
property. We discuss properties of big Cohen-Macaulay module closures, and
prove that every Dietz closure is contained in a big Cohen-Macaulay module
closure. Using this result, we show that under mild conditions, a ring R is
regular if and only if all Dietz closures on R are trivial. Finally, we show
that solid closure in equal characteristic 0, integral closure, and regular
closure are not Dietz closures, and that all Dietz closures are contained in
liftable integral closure.
