Hereditariness Of Direct Sum Decomposition To Direct Summand And Cancelation Of Modules

The theory of direct decomposition of modules has been considered as both central subjects in module theory and important tools in ring theory.Moreover many branches have been developed deeply in this area.Since the well known Krull-Schmidt Theorem concerning the module with finite length appeared in 1932,the problems on uniqueness of direct decomposition have been studied widely in order to solve the problems pointed by Krull.Especially there appeared many new concepts and problems comprising the exchange property by Crawlay and Jonssonin in 1964,the concept of decompositions complement direct summands, the problems pointed by Warfield in 1975,the theory of factor categories by Harada in the 1980s and the properties of commutative semigroups in the categories of modules in recent years,all of which greatly promoted direct decomposition theory.Furthermore,Facchini introduced a kind of corresponding relation (similar to "bijection" between two sets)between a monogeny class and an epigeny class in the course of dealing with the problem by Warfield, and his conclusions extended the signification of uniqueness of direct decomposition.Brookfield proved that there was an isomorphism map between two essential submodules from different direct summands,and then gave the meanings of "cancellation" from a new point of view. However in order to solve the problems above,it has became a popular way to deal with some special categories of modules in recent years.This dissertation is devoted two topics on direct decomposition of modules;cancellation and hereditariness of direct sum decompositions to direct summand.Cancellation is an important branch of the theory of direct decomposition of modules,and has connections with many other areas,for example its application to compute the K₀ group of a ring. On the other hand,hereditariness of direct sum decompositions to direct summand can be seen as the generalization of the well-known Kaplansky Theorem.Firstly,the second and the third open problems pointed by Facchini are considered in the course of dealing with hereditariness.For the second open problem,we present a few new sufficient conditions and a necessary condition making hereditariness true.At the same time,we give the concept of isomorphic complement(maximal)direct summands whose condition is weaker than one of complement(maximal)direct summands. And then we point out the difference between the concept of isomorphic complement(maximal)direct summands and the concept of complement (maximal)direct summands,and then obtain a property of decomposi-tion with isomorphic complement(maximal)direct summands.For the third open problem(a special case of the second open problem),we ob-tain a sufficient and necessary condition of the result that the direct sum of indecomposable injective modules is an injective module by introduc-ing the condition(A)of a module M,and then give a sufficient condition making the third open problem true.Secondly,we improve two properties concerning semiexchange prop- erty with respect to,and give the concept of -semiexchange ring.Fur-thermore,we character the relation between -semiexchange ring and its endomorphism ring.At last,we give a new category in order to deal with cancellations, and generalize the Brookfield's conclusion concerning modules.For finite Goldie dimension,The analogical conclusion on modules is obtained.For Evans Theorem,Krull-Schmidt Theorem and other classical conclusion in module theory,their analogue in this new category are also obtained respectively.The results obtained are different from cancellation in the usual sense,but it can be seen cancellation in a new sense.The main results of this dissertation are as followsTheorem 2.1.2.Let(M_α)_α∈Abe an indexed set of modules such that End(M_α)is a local ring for everyα∈A,and let M=⊕_AM_α.Assume thatA=U_i∈IA_i such that A_i is a finite set for every i∈IandA_i∩A_j=φfor any i≠j∈I.Then for a summand N of M,there exists a decompositionN=⊕_BN_βsuch that B(?)A and N_β(?)M_βfor allβ∈B in case that one of the following holds;(1)For anyα∈A_i,β∈A_j such that i≠j∈I,M_αand M_βare homologically independent;(2)M_αand M_βare isomorphism if and only ifα,β∈A_i for some i∈I ,and any submodule ofM which is isomorphism to some summand of M is also a summand of M.Theorem 2.1.5.Let(M_α)_α∈Abe independent submodules of a module M such that M=⊕_AM_αis an indecomposable decomposition,and is isomorphic complement maximal direct summands,that is M=N₁⊕…⊕N_m⊕D,where each End(N_i)is a local ring.Then there exists a subset {i₁,…,i_m}(?)A,and there exist submodules P_i1,'",P_imof M such that P_ik(?)M_ikand M=P_i1⊕…⊕P_im⊕D. Theorem 2.1.6.Let(M_α)_α∈Abe an indexed set of modules such that End(M_α)is a local ring for anyα∈A,and let M=⊕_AM_α.Assume that M=P⊕Q.If one of the following holds,then has an indecomposable decomposition.(a)Either P or Q has exchange property;(b)Q has an indecomposable decomposition,and M satisfies internal cancellation.Theorem 2.1.10.Let M=⊕_AM_αbe an indecomposable decomposition with complement maximal direct summands such that any summand of M has an indecomposable decomposition.Then for any summand P of M,there exists a indecomposable summand in Q,and the modules class that is locally isomorphism to direct summands in P is isomorphism to direct summands.Theorem 2.2.2.Let M=N^(I),where N is an indecomposable injective module,and I is an infinite index set.Then the following conditions are equivalent;1)M is injective;2)For any nonempty set A,N^(A)is a summand of N^A;3)For a denumerable set K,N^(K)is a summand of N^K;4)N is∑-algebraically compact.Theorem 2.2.4.Let M=⊕_AN_α,where N_αis an indecomposable injective module for anyα∈A.Then M is injective if and only if M satisfies the condition(A).Theorem 2.3.3.Let M be a finitely generated module.Then M has the semiexchange property with respect to m if and only if M has the finite semiexchange property with respect to m.Theorem 2.3.5.Let R be an m -semiexchange ring.If R does not contain infinite orthogonal idempotents,then R is semilocal.Theorem 2.3.6.Let M be a R-module and S=End(M).Then M has the finite semiexchange property with respect to m if and only if S is m-semiexchange ring.Theorem 3.2.1.Assume that A⊕C(?)B⊕C,where C has finite Goldie dimension.Then there exist essential submodules A′(?)A and B′(?)B such that A′(?)B′.Theorem 3.2.2.Let M be a module with Goldie dimension equaling to n.If M has two indecomposable decompositions M=A₁⊕…⊕A_k=B₁⊕…⊕B_m, then k=m=n and A_i(?)B_σ(i)for some permutationσ.Theorem 3.2.3.Let B be a module with finite Goldie dimension and Aⁿ(?)Bⁿ.Then there exist essential submodules A′(?)A and B′(?)B such that A′(?)B′.