Topological and algebraic theory of convex subrings of *R

In this project, we investigate some topological and algebraic structures of convex subrings of *R, a nonstandard extension of the field of real numbers *R. In particular, a detailed description of the prime spectrum of any convex subring of *R is given. Moreover, associated to any convex subring of*R, two monoids of naturals are studied in detail. There appeared to be an effective link between these two sets of naturals and bounded polynomials. In fact, by taking the standard part mapping, a bounded polynomial gives an entire holomorphic function.