形式矩阵环上的模，映射和零因子图的研究

发布时间：2018-05-14 05:31

本文选题：形式矩阵环 + 模　；参考：《广西师范学院》2017年硕士论文

【摘要】：形式矩阵环是矩阵环的推广,它在环论和模论中都起着重要的作用.众所周知,每个具有非平凡幂等元的环都与一个形式矩阵环同构,每个可分解模的自同态环也与一个形式矩阵环同构.形式三角矩阵环是一类重要的形式矩阵环,它在Artin代数的表示中起着重要作用.形式矩阵环具有丰富的性质和重要的应用,对环的研究具有重要意义.本文在前人的基础上进一步研究了形式矩阵环上的模,映射和零因子图.第一章介绍了本文的研究背景,研究意义,以及本文涉及到的一些基本概念和相关结论.第二章主要研究了形式矩阵环(?)上的artinian模,noetherian模和有限表现模.证明了任意右K -模(X,Y)f,g是右artinian (noetherian)模当且仅当右R-模X和右S -模Y是artinian (noetherian)模.给出了形式矩阵环K上的模是有限表现模的充分条件.此外还研究了任意右K-模的多余满同态,证明了任意右K -模(X,Y)f,g有投射覆盖当且仅当R-模X / YN和S -模Y/XM有投射覆盖.第三章主要研究了具有零迹理想的形式矩阵环的环同态,σ-导子,σ-双导子和σ-交换映射.证明了在一定条件下σ-双导子是外σ-双导子与内σ-双导子的和并得到了σ-双导子是内σ-双导子的充分条件.给出形式矩阵环上的σ-交换映射的具体形式,得出了σ-交换映射是真的σ-交换映射的一些等价刻画并给出σ-交换映射是真的σ -交换映射的一个充分条件.第四章主要研究了交换环上的n阶形式矩阵环的零因子及零因子图的性质.引入了环上左(右)形式线性方程组的概念,并用其证明了M_n(R;S_(ijk)})的元素A是零因子当且仅当它的行列式是R的零因子当且仅当A是R[A]的零因子.刻画了交换环R上的形式矩阵环M_n(R;{S_(ijk)})的无向零因子图Γ(M_n(R;{S_(ijk)}))和有向零因子图Γ(M_n(R;{S_(ijk)})).证得Γ(M_n(R;{S_(ijk)}))是非平面图,围长都是3,直径只能是2或3.还证明了有向零因子图Γ(M_n(R;{S_(ijk)}))的直径也只能是2或3且Γ(M_n(R;{S_(ijk)}))(?)Γ(M_n(T(R);{S_(ijk)})),其中T(R)是交换环R的全商环,
[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.
【学位授予单位】：广西师范学院
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O153.3

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