On Global C-Dimensions And Coherent Properties Of FI-Modules

In this thesis,we concern global C-dimensions and coherent properties of FIG-modules.Let R be a ring with 1 and M the left or right R-module category.Let C be the class of R-modules closed under extensions,direct summands and isomorphisms.Suppose C is both precovering and preenveloping.In chapter two,we prove that the mth C-cosyzygy of any R-module is contained in C if and only if the mth C-syzygy of any R-module is contained in C;if C is closed under cokernels of monomorphisms,then gl right C-dim M≤m and both the mth and (m+1)th C-cosyzygy of any R-module have a monic C-preenvelope if and only if gl left C-dim M≤m-2(m≥2);if C is closed under kernels of epimorphisms,then gl left C-dim M≤m and both the mth and (m+1)th C-syzygy of any R-module have an epic C-precover if and only if gl right C-dim M≤m-2(m≥2);if every nonzero R-module has a nonzero C-preenvelope and a nonzero C-precover,then gl right C-dim M=gl left C-dim M.In chapter three,we illust.rate the main results in the preceding chapter by considering certain categories of modules.In particular,some results obtained by Prof.Enochs or Lixin Mao and Nanqing Ding are improved.New cotorsion pairs are discovered.In the final chapter,we consider an open problem posted by Sam and Snowden in 2014 when the category in question is FIG.We prove that,over a commutative coherent ring,every finitely presented FIG-module is coherent.It turns out that many properties of finitely presented FIG-modules may be generalized from Noetherian rings to coherent rings.