Properties And Applications Of Some Precovers

The exsitence of (pre)cover and (pre)envelope is a classical problem in ring theory. (pre)Covcr and (pre)envelope with different properties can characterize differentkinds of rings. Auslander defines it in algebra with the applications of almost split, sequence , now it. is used widely in the representation theory of Artin algebra. This paper is mainly divided into two parts. One is the properties of special preenvelopesand special precovers, with their existences and their characterizations of each other, their preservation under certain functors. Then we discuss the propertiesof Gorenstein almost precover. In the second part, we gave an application of special (?)-precover. and get some relative homological results. Then we discussd the relations between (?)-dimension with projective dimension. Main results:Theorem 1.2.1. X is closed under direct limits. X_i∈χ, If f_i : X_i→M_i is a specialχ- precover, thenΠf_i:ΠX_i→ΠM_i is a specialχ- precover.Theorem 1.2.3. If 0→A→B→C→0 is a short exact sequence. If either of A, B. C has a special precover. suppose is X₁, X₂, X₃. Then we have the following diagram commutes: Theorem 1.2.8. As defined in Definition 1.2.5, if F is a separable functor, if every object in (?) has a special (?)-precover, then every object in (?) has a special (?)-precover.Theorem 1.3.3. R is right strongly coherent ring, and injective left R-module is local protective, then every R-module has a special ^⊥(LP)-precover.Theorem 1.3.10. On n-Gorenstein ring, module M has a special ((?)F)-preenvelope if and only if M has a special ^⊥(((?)F)^⊥)-precover.Lemma 1.4.6. Consider the following pullback diagram:(1)Tf g is Gorenstein essential monomorphism,f is monomorphism, thenαis Gorensteinessential monomorphism.(2)If both f and g are Gorenstein essential monomorphisms, thenαandβare Gorenstein essential monomorphisms.Theorem1.4.10. If (?) is Gorenstein weakly hereditary category. If (?)i : X_i→M_i,i = 1.2,3, ...,n. is Gorenstein almost (?)-precover. Then (?)(?)i : (?)X_i→M_i is Gorensteiii almost (?)-precover. Theorem 2.1.7. If f : M→M' is a module morphism, consider the followingdiagram:the upper row is (?)- decomposition of M, the lower row is (?)- decomposition of M . Then there exists a chain morphism F : X_M→X_M' making the following diagram commutes, and any two homomorphisms satisfying the conditions are homotopy.Theorem 2.1.8. For any abclian covariant functor T, the left derived functor L_nT and L'_n T are naturally equivalent. Specially, for any M∈R- Mod. we have(L_nT)M(?)(L'_nT)M．Theorem 2.2.2. If the (?)d of M is m, then pd_RM≤m + n.Crolaray 2.2.3. For any ring R, and any left R-module M, the following conditions arc equvalent:(1)pdM<∝,(2)(?)dM<∝．Crolaray 2.2 .4. For any short exact sequence 0→M₁→M→M₂→0. if any two of (?)dM₁,(?)dM, (?)dM₂ is finite, then the third is finite.