When R(X) and R⟨X⟩ are Clean: A Constructive Treatment

Abstract

For R a commutative ring, we give constructive proofs that R(X) is clean exactly when R is clean, and that R ⟨ X ⟩ is clean exactly when R is zero dimensional. We also give a constructive proof of the known result that R(X) = R ⟨ X ⟩ 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.