  We prove well-posedness and regularity for the stochastic pure Cahn-Hilliard
equation under homogeneous Neumann boundary conditions, with both additive and
multiplicative Wiener noise. In contrast with great part of the literature, the
double-well potential is treated as generally as possible, its convex part
being associated to a multivalued maximal monotone graph everywhere defined on
the real line on which no growth nor smoothness assumptions are assumed. The
regularity result allows to give appropriate sense to the chemical potential
and to write a natural variational formulation of the problem. The proofs are
based on suitable monotonicity and compactness arguments in a generalized
variational framework.
