  The first aim of this paper is to examine existence, uniqueness and
regularity for the Cahn-Hilliard-Cook (CHC) equation in space dimension $d\leq
3$. By applying a spectral Galerkin method to the infinite dimensional
equation, we elaborate the well-posedness and regularity of the finite
dimensional approximate problem. The key idea lies in transforming the
stochastic problem {\color{black}{with additive noise}} into an equivalent
random equation. The regularity of the solution to the equivalent random
equation is obtained, in one dimension, with the aid of the Gagliardo-Nirenberg
inequality and done in two and three dimensions, by the energy argument.
Further, the approximate solution is shown to be strongly convergent to the
unique mild solution of the original CHC equation, whose spatio-temporal
regularity can be attained by similar arguments. In addition, a fully discrete
approximation of such problem is investigated, performed by the spectral
Galerkin method in space and the backward Euler method in time. The previously
obtained regularity results of the problem help us to identify strong
convergence rates of the fully discrete scheme.
