Added: as it was kindly noted, it happens that the ring of continuous functions (on a generic completely regular space) is not a classical ring of quotients; its ring of quotients is obtained considering functions which are definite and continuous on a dense co-zero set (dense open set in the normal case), two such functions being identified when they coincide on a dense co-zero set. But then this ring of quotients is von Neumann regular (and it isnaturally identified with the ring of continuous functions on a certain almost P-spaceassociated to the original space); being von Neumann regular there is no hope to obtain the kind of example wanted (as it was implicitly remarked by the requester). However,the idea that "regular" functions can be modified on a ideal of "small" sets (in this case, the sets whose closure has empty interior) still works, and in another answer it waswell used (with polynomial functions as regular functions and finite sets as ideal of small sets. One can use other choices of regular functions and/or small sets: meager sets,measure zero sets, ... however, for algebraic geometry, polynomials and finite setsare the most natural choices)
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[I might have misunderstood your equivalent question]. Some hints: a commutative reduced ring is the same as a subring of a direct product of fields (i.e. a ring of field-valued functions, but the co-domain can change with the point); you also wants that never zero functions have an inverse. So typically one considers continuous, of differentiable (or whatever) real valued functions. Then you want to invert also a function f that is sometimes zero; this "kills" the zero set Z of f (the localization is a ring of functions defined outside Z). But then you want that every g which is nonzero outside Z has a inverse, i.e. inverting one particular f should invert every g. Taking the sequences which have limit (i.e. the continuous functions on the Alexandroff compactification of the naturals), suppose that you want to invert f(n)=1/n ; does this invert all positive and converging to zero sequences? Inverting f gives functions that are of polynomial growth when n tends to infinity, so what happens for a g with exponential decrease? |
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