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We continue the investigation of the algebraic and topological structure of the algebra of Colombeau generalized functions with the aim of building up the algebraic basis for the theory of these functions. This was started in a previous work of Aragona and Juriaans, where the algebraic and topological structure of the Colombeau generalized numbers were studied. Here, among other important things, we determine completely the minimal primes ofK and introduce several invariants of the ideals of G(Ω). The main tools we use are the algebraic results obtained by Aragona and Juriaans and the theory of differential calculus on generalized manifolds developed by Aragona and co-workers. The main achievement of the differential calculus is that all classical objects, such as distributions, become C ∞ -functions. Our purpose is to build an independent and intrinsic theory for Colombeau generalized functions and place them in a wider context.

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144:13-29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144:13-29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green's theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.

We define intrinsic, natural and metrizable topologies TΩ, T , Ts,Ω and Ts in G (Ω), I K, Gs(Ω) and I Ks respectively. The topology TΩ induces T , Ts,Ω and Ts. The topologies Ts,Ω and Ts coincide with the Scarpalezos sharp topologies. * 2000 Mathematics Subject Classification: Primary 46F30 Secondary 46T20.