Homological Character Of Min-injective Modules

In this thesis, we mainly discuss the homological character of min-injective mod-ules.Injective module is the important part of homological algebra.In recent years,it has been studied by many authors(e.g,W.K.Nicholson , M.F.Yousif , Chenjianlong and Dingnanqing,etc.)In the second chapter, we mainly discuss the fundamental character of min-injective modules.Using min-injective modules depict P.S ring.Afterwards,we discuss the endmorphism ring of quasi-min-injective modules, obtaining some equivalent condition .Main results followTheorem2.1.7 Let R be a ring and M a right min injective module,N≤ M, π : M → M/N epimorphism, then the following are equivalent:(1)N is min-injective module(2)For every simple ideal A of R, g ∈ Hom_R(R/A, M/N),there have h ∈ Hom_R(R/A,M), such thatg = πh(3)N{A) = N + rM(A)for N(A) = {x ∈ M|Ax ? TV}, simple ideal A of R.Theorem 2.2.1 Let R be a ring,then the following are equivalent:(1) R is P.S-ring;(2)Every quotion module of min-injective module is min-injective module;(3)Every quotion module of min-injective module is min-injective module;(4)Mis injective module,f ∈ End_R(M), then/m/ is min-injective.In the third chapter,we definite min-injective dimension of module and ring. We obtain the relation between min-injective dimension and weak dimension,global dimension. More over,Using min-injective dimension of ring class rings.Main resultsfollow. letK be sim-coherent rings,then R is/F ring if and only ifR is right min-injective ring.In the forth chapter,we definite min-projective dimension of module and ring.we obtain some fundamental character.Main results follow:Theorem4.2.Let R be sim-coherent ring,then the following are equavelent:(l)mpD(R) < 1;(2)Submodule of min-projective module is min-projective module;(3)Submodule of projective module is min-projective module;(4)Each right ideal of Ris min-projective.