U-Matlis Cotorsion Modules And Module-Theoretic Characterization Of G-domains

In this thesis,we mainly discuss the related properties of u-Matlis cotorsion modules and give module-theoretic characterization of G-domains.we proved that G-domains is Matlis domains.We study the u-divisible modules,u-injective modules,u-flat modules.Let R be a commutative ring and u ∈ R be a non-zero divisor.Let M be R-module.If ExtR1(Ru,M)=0,then M is called an u-Matlis cotorsion module.If L)=0,then L is called an u-divisible module.An ideal I is said to be an u-ideal if I∩un ≠(?).A module E is said to be an u-injective module if ExtR1(R/I,E)=0,for any u-ideal I of R.A module N is said to be an u-flat module if Tor1R(N,R/I)=0.We proved that every u-divisible module is a quotient module of u-injective module.If every u-injective module is FP-injective module,then u-flat module is flat.Moreover,we prove that R is a G-domain if and only if every u-injective module is injective,if and only if every u-Matlis cotorsion module is Matlis cotorsion.