Algebraic theory of Colombeauʼs generalized numbers

Let denote the commutative ring of Colombeauʼs full generalized numbers. Endowed with Scarpalezosʼ sharp topology it becomes a topological ring. We study the algebraic and topological properties of this topological ring. In particular, we prove that the group of units of is dense in the sharp topology, determine its boolean algebra, show that it has minimal primes, describe them completely which results in a complete classification of the maximal ideals. From the description of the prime and maximal ideals, it becomes clear that they should be determined by certain ultra-filters.