Algebraic theory of Colombeauʼs generalized numbers

Abstract

Let K¯ 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 K¯ 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.