On the prime spectrum of the ring of bounded nonstandard complex numbers

In this paper, we provide some algebraic structures of convex subrings of * C, a nonstandard extension of the field of complex numbers C. In particular, a detailed description of the prime spectrum of any convex subring of * C is given. To achieve our goal, first we investigate prime ideals and we characterize two consecutive elements in the spectrum of a divided domain. We also show that the prime spectrum of the ring of bounded hypercomplex numbers has two peculiar properties: there are no three consecutive elements in the spectrum; moreover, nonzero elements are a disjoint union of three subsets where one of them is strongly dense and the other two are dense in the spectrum.