  We develop the basic theory of projective modules and splitting in the more
general setting of systems. Systems provide a common language for most tropical
algebraic approaches including supertropical algebra, hyperrings (specifically
hyperfields), and fuzzy rings. This enables us to prove analogues of classical
theorems for tropical and hyperring theory in a unified way. In this context we
prove a Dual Basis Lemma and versions of Schanuel's Lemma.
