Reverse Mathematics, Projective Modules and Invertible Modules

摘要

We study projective modules and invertible modules by techniques of reverse mathematics. Dual Basis Lemma provides an equivalent characterization of projective R-modules M via their dual Hom_R(M, R). It is a useful tool to prove various theorems about projective modules. We first formalize and prove the Dual Basis Lemma in RCAo. Then we study Kaplansky's Theorem, which says that every submodule of a free module over a hereditary ring is projective. We show that RCAo proves that a submodule of a free module over a Σ_1~0-hereditary ring is a direct sum of projective modules, and thus projective. By defining invertible R-submodules of an extension ring of R via Σ_2~0 formulas, we show that RCA_0 proves the statement that invertible R-modules are finitely generated projective R-modules. Modified Projectivity Test and Modified Injectivity Test are basic tests for determining projective modules and injective modules, respectively. Lastly, we show that the Modified Projectivity Test and the Modified Injectivity Test are provable in ACA_0 and RCA_0, respectively.