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Several types of rings of smooth functions, such as differentiable algebras and formal algebras, occupy a central position in singularity theory and related subjects. In this series of papers we will be concerned with a larger class of rings of smooth functions, which would play a role in Differential Geometry similar to the role played by commutative rings or k-algebras in Algebraic Geometry. This larger class of rings is obtained from rings of smooth functions on manifolds by dividing by ideals and taking filtered colimits. The original motivation to introduce and study P-rings was to construct topos-models for synthetic differential geometry (SDG). The program of SDG (see, e.g., Kock [ 11) was proposed by F. W. Lawvere, and it was in this context that C-rings first appeared explicitly in the literature (see, e.g., Reyes and Wraith [14] and Dubuc [2]). These toposes which provide models for SDG are constructed in a way similar to the toposes occurring in algebraic geometry, but with k-algebras replaced by C"-rings. In particular, the C"-analogue of the Zariski topos, the so-called smooth Zariski topos of Moerdijk and Reyes [ 111 contains a category of "smooth schemes," just as the usual Zariski topos contains the schemes of algebraic geometry (see Demazure and Gabriel [ 11). Despite this original motivation from SDG, P-rings and their schemes can be studied by themselves, and independently from topos theory in general, and topos-models for SDG in particular. In these two papers, we will start to explore this independent line of development of the theory of P-rings. This can make the connection with algebraic geometry stronger, since the usual presentation of the relation between algebra and geometry takes place at the level of schemes, rather than toposes. The organization of this paper and its sequel, part 11, is as follows. 324

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