Construction of a maximal ideal
Let R denote the ring of continuous functions defined on the real line, let I in R be the ideal consisting of functions with compact support. Obviously, I is not maximal, and by Zorn's Lemma there exists a maximal ideal M in R which contains I. Is there now an explicit construction or characterization for M, i.e. when given any function in R, can one decide whether it lies in M or not?