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We give new proofs of two results of Stafford, which generalize two famousTheorems of Serre and Bass regarding projective modules. Our techniques areinspired by the theory of basic elements. Using these methods we furthergeneralize Serre's Splitting Theorem by imposing a condition to the splittingmaps, which has an application to the case of Cartier algebras.

Generalizing Serre's Splitting Theorem and Bass's Cancellation Theorem via free-basic elements