For R a commutative ring, we give constructive proofs that R(X) is clean exactly when R is clean, and that R is clean exactly when R is zero dimensional. We also give a constructive proof of the known result that R(X) = R exactly when R is zero dimensional. By a constructive proof we mean one that is carried out within the context of intuitionistic logic. In practice, this means that the arguments are arithmetic rather than ideal theoretic.
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