The discrete module category for the ring of K-theory operations

We study the category of discrete modules over the ring of degree-zero stable operations in $p$-local complex $K$-theory, where $p$ is an odd prime. We show that the $K_{\left(p\right)}$-homology of any space or spectrum is such a module, and that this category is isomorphic to a category defined by Bousfield and used in his work on the $K_{\left(p\right)}$-local stable homotopy category. We give a simple construction of cofree discrete modules and construct the analogue in the category of discrete modules of a four-term exact sequence due to Bousfield.