本文選題：形式矩陣環(huán) + 模��； 參考：《廣西師范學院》2017年碩士論文

【摘要】：形式矩陣環(huán)是矩陣環(huán)的推廣,它在環(huán)論和模論中都起著重要的作用.眾所周知,每個具有非平凡冪等元的環(huán)都與一個形式矩陣環(huán)同構,每個可分解模的自同態(tài)環(huán)也與一個形式矩陣環(huán)同構.形式三角矩陣環(huán)是一類重要的形式矩陣環(huán),它在Artin代數(shù)的表示中起著重要作用.形式矩陣環(huán)具有豐富的性質和重要的應用,對環(huán)的研究具有重要意義.本文在前人的基礎上進一步研究了形式矩陣環(huán)上的模,映射和零因子圖.第一章介紹了本文的研究背景,研究意義,以及本文涉及到的一些基本概念和相關結論.第二章主要研究了形式矩陣環(huán)(?)上的artinian模,noetherian模和有限表現(xiàn)模.證明了任意右K -模(X,Y)f,g是右artinian (noetherian)模當且僅當右R-模X和右S -模Y是artinian (noetherian)模.給出了形式矩陣環(huán)K上的模是有限表現(xiàn)模的充分條件.此外還研究了任意右K-模的多余滿同態(tài),證明了任意右K -模(X,Y)f,g有投射覆蓋當且僅當R-模X / YN和S -模Y/XM有投射覆蓋.第三章主要研究了具有零跡理想的形式矩陣環(huán)的環(huán)同態(tài),σ-導子,σ-雙導子和σ-交換映射.證明了在一定條件下σ-雙導子是外σ-雙導子與內σ-雙導子的和并得到了σ-雙導子是內σ-雙導子的充分條件.給出形式矩陣環(huán)上的σ-交換映射的具體形式,得出了σ-交換映射是真的σ-交換映射的一些等價刻畫并給出σ-交換映射是真的σ -交換映射的一個充分條件.第四章主要研究了交換環(huán)上的n階形式矩陣環(huán)的零因子及零因子圖的性質.引入了環(huán)上左(右)形式線性方程組的概念,并用其證明了M_n(R;S_(ijk)})的元素A是零因子當且僅當它的行列式是R的零因子當且僅當A是R[A]的零因子.刻畫了交換環(huán)R上的形式矩陣環(huán)M_n(R;{S_(ijk)})的無向零因子圖Γ(M_n(R;{S_(ijk)}))和有向零因子圖Γ(M_n(R;{S_(ijk)})).證得Γ(M_n(R;{S_(ijk)}))是非平面圖,圍長都是3,直徑只能是2或3.還證明了有向零因子圖Γ(M_n(R;{S_(ijk)}))的直徑也只能是2或3且Γ(M_n(R;{S_(ijk)}))(?)Γ(M_n(T(R);{S_(ijk)})),其中T(R)是交換環(huán)R的全商環(huán),
[Abstract]:Formal matrix ring is a generalization of matrix ring, which plays an important role in ring theory and module theory. It is well known that every ring with nontrivial idempotent elements is isomorphic to a formal matrix ring, and each endomorphism ring of decomposable modules is also isomorphic to a formal matrix ring. Formal triangular matrix ring is an important class of formal matrix ring, which plays an important role in the representation of Artin algebra. Formal matrix rings have rich properties and important applications, which is of great significance to the study of rings. In this paper, we further study the modules, mappings and zero digraphs over formal matrix rings on the basis of previous studies. The first chapter introduces the research background, research significance, and some basic concepts and related conclusions. In the second chapter, we mainly study the formal matrix ring. Artinian modules and finite representation modules. In this paper, we prove that any right K-module X _ T _ y _ F _ G is a right artinian not etherian) module if and only if the right R-module X and the right S-module Y are artinian noetherian) modules. The sufficient conditions under which the modules over the formal matrix ring K are finite representation modules are given. In addition, the superfluous full homomorphisms of any right K-module are studied, and it is proved that any right K-module X ~ (+) y ~ (+) F _ (G) has projective covering if and only if the R _ (-) -module X / YN and S-module Y/XM have projective covers. In chapter 3, we study the ring homomorphism, 蟽 -derivation, 蟽 -biderivation and 蟽 -commutative mapping of formal matrix rings with zero trace ideals. It is proved that 蟽 -biderivation is the sum of outer 蟽 -biderivation and inner 蟽 -biderivation under certain conditions, and the sufficient condition that 蟽 -biderivation is internal 蟽 -biderivation is obtained. This paper gives the concrete form of 蟽 -commutative mappings over formal matrix rings, obtains some equivalent characterizations that 蟽 -commutative mappings are true 蟽 -commutative mappings, and gives a sufficient condition that 蟽 -commutative mappings are true 蟽 -commutative mappings. In chapter 4, we study the properties of zero divisor and zero digraph of n order formal matrix rings over commutative rings. In this paper, the concept of left (right) form linear equations over rings is introduced, by which it is proved that the element A of M _ nn / R _ T _ S _ I _ j _ k}) is zero if and only if its determinant is the zero factor of R if and only if A is the zero factor of R [A]. In this paper, we describe the undirected zero-divisor graph 螕 / M _ nn _ (R) and directed zero _ factor graph 螕 ~ (M ~ n ~ r; {S _ S _ T _ I _ j _ k)} over a commutative ring R. ({S _ S _ T _ I _ j _ k}) and a directed zero _ factor graph 螕 ~ (?) ~ M _ n _ n ~ r; {S _ S _ S _ I _ j _ k}. The results show that 螕 / M / M / M / T ({S / S / T _ I _ j _ k)} is a displanar graph with a girth of 3 and a diameter of only 2 or 3. It has also been proved that the directed zero-factor graph 螕 / S / T / M / T / R; {S / S / S / T _ j _ k}) can only be 2 or 3 in diameter and 螕 / C / M / M / N / R; {S / S / S / S / T _ k} / T / S / S / S / S / S / S / S / T / T / S / S / S / S / S / T / T / T / T / T / T) and that the T _ T _ R) is the total quotient ring of the exchange ring R.
【學位授予單位】：廣西師范學院
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O153.3

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