  We study closure operations over a local domain R that satisfy a set of
axioms introduced by Geoffrey Dietz. The existence of a closure operation
satisfying the axioms (called a Dietz closure) is equivalent to the existence
of a big Cohen-Macaulay module for R. We give an additional axiom (the Algebra
Axiom), such that the existence of a Dietz closure satisfying this axiom is
equivalent to the existence of a big Cohen-Macaulay algebra. We prove that many
closure operations satisfy the Algebra Axiom, whether or not they are Dietz
closures. We discuss the smallest big Cohen-Macaulay algebra closure on a given
ring, and show that every Dietz closure satisfying the Algebra Axiom is
contained in a big Cohen-Macaulay algebra closure. This leads to proofs that in
rings of characteristic p > 0, every Dietz closure satisfying the Algebra Axiom
is contained in tight closure, and there exist Dietz closures that do not
satisfy the Algebra Axiom. Several of these results rely on work from
arXiv:1512.06843.
