WebJul 5, 2016 · 1Free modules 2Projective modules 3Flat modules 4Torsion-free modules Free modules[edit edit source] The following definitions are straightforward … WebFeb 14, 2024 · Skew-fields can be characterized as rings over which all modules are free. Over a principal ideal domain a submodule of a free module is free. Near to free modules are projective modules and flat modules (cf. Projective module; Flat module). References
Projective Module -- from Wolfram MathWorld
Webis a free -module, and is a projective -module. Proof. Since any projective module is flat (as a direct summand of a free module) and every free module is projective, it suffices to prove that a flat module is free. Let be a flat module. Let be a set and let , be elements such that forms a basis over the residue field of . WebIn particular, every module has free resolutions, projective resolutions and flat resolutions, which are left resolutions consisting, respectively of free modules, projective modules or flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules . Resolutions of modules [ edit] glutamine for alcohol recovery
Free modules, finitely-generated modules - University of …
Web1) Free modules are of course an easy sell: for such things the usual notions of linear algebra work well, including that of dimension (called "rank" in this case: note that my rings are commutative!). WebApr 19, 2024 · A free module is flat. More generally, a projective module is flat. Proof Since A is flat, by proposition 1, any free module is also flat. If P is projective, is free for some A -module Q. Thus is flat. By proposition 1 again, P is flat. ♦ Proposition 3. If B is a flat A-algebra and M is a flat B-module, then M is a flat A-module. WebAn -module is called flat if whenever is an exact sequence of -modules the sequence is exact as well. An -module is called faithfully flat if the complex of -modules is exact if and only if the sequence is exact. A ring map is called flat if is flat as an -module. A ring map is called faithfully flat if is faithfully flat as an -module. bok 71st and sheridan