Closed exact categories of modules over generalized adic rings. Part 1: The bounded case

We develop general foundations of topological algebra over a linearly topologized ring k in a format applicable to both formal schemes and analytic adic spaces. We are especially interested in determining exact closed tensor categories of complete linearly topologized k-modules, with enough projectives or injectives. For k a widely generalized adic ring, we describe here a few examples of such categories consisting of bounded modules. The application to the construction of quasi-coherent modules over formal schemes will be given elsewhere.