Abstract
A ring extension is a ring homomorphism preserving identities. In this paper, we give the definition of relatively free modules on ring extensions and develop some basic properties of relatively free modules. Then we establish the relationship between relatively free modules and relatively projective modules. In particular, we prove that the relatively free modules, relatively projective modules and relatively injective modules on the ring extension coincide with being a natural number, and that every such module has the form or with N an S-module.
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