  The methods of nonstandard analysis are applied to algebra and number theory.
We study nonstandard Dedekind rings, for example an ultraproduct of the ring of
integers of a number field. Such rings possess a rich structure and have
interesting relations to standard Dedekind rings and their completions. We use
lattice theory to classify the ideals of nonstandard Dedekind rings. The
nonzero prime ideals are contained in exactly one maximal ideal and can be
described using valuation theory. The localisation at a prime ideal gives a
valuation ring and we determine the value group and the residue field. The
spectrum of a nonstandard Dedekind ring is described using lattices and value
groups. Furthermore, we investigate the Riemann-Zariski space and the valuation
spectrum of nonstandard Dedekind rings and their quotient fields.
