环左理想半群的Cayley图特性研究（2014）

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本文档探讨了环R与左理想半群IL(R)的Cayley图Cay(L(R))之间的相互作用。Cayley图是一种基于代数结构构建的图形，它在图论中有着广泛的应用，尤其是在表示群和半群的元素结构时。本文主要关注的是对这个特定半群Cayley图的图论性质进行深入研究，包括但不限于： 1. **Transitive closure**：作者可能考察了图Cay(L(R))是否具有传递闭包性质，即任意两个左理想通过一系列的连接可以达到，这对于理解环R的结构以及其左理想的组合关系至关重要。 2. **Girth**：Girth是指图中长度最短的简单回路。如果Cay(L(R))有确定的girth，那么这将揭示环R的某种局部特征，如是否存在特殊的环面结构或循环子结构。 3. **Radius** 和 **Diameter**：这两个概念分别表示从图中任意一个点到其他所有点的最短距离和最长距离。它们提供了关于环R左理想集合中信息传播速度和结构紧凑性的度量。 4. **Semi-simplicity and spanning subgraphs**：作者可能讨论了如何识别那些包含在Cay(L(R))中的子图，这些子图能够作为图的生成子图，即任何两点都能通过它们之间的路径相连，从而对环R的简化表示提供一种有效的途径。 5. **Regularity conditions**：文章还可能探讨了特定条件，当满足这些条件时，环R会显示为正规环（所有元素都是正规元）、左二重环（左乘与右乘的结合律成立）或者其幂等元（满足x*x=x的元素）集中在中心。 6. **Characterization of rings and ideals**：作者可能通过Cayley图Cay(IL(R))来刻画不同类型的环，如通过子图特性揭示哪些环是简单环，或者如何通过图的结构来确定特定的左理想特性。关键词：“Cayley graph”、“semigroup of left ideals”、“ring theory”表明本文的核心是运用图论工具来分析环论中的左理想结构，这对于理解抽象代数中的环结构及其应用具有重要意义。通过深入研究这些图形性质，作者旨在揭示环R的内在结构和特性，同时也在探索图论与代数之间的联系。

VoL 43 ,No. 5 Journal of Shanghai Normal University( Natural Sciences · Mathematics)

DOI :10. 3969/J. ISSN. 100 -5137. 2014.05.007

Oct.， 2014

The Cayley graph built upon the semigroup of

left ideals of a ring

Ailuia Li1, Ralph P. rFucci2

(I. Department of Mathematical Sciences, Montclair State University, Montclair, New Jersey 07043, USA

；

2. Department of Mathematical Sciences, Loyola University New Orleans, New Orleans, Louisiana 70118, USA)

Abstract： We investigate the interaction between a ring R and the Cayley graph Cay(JL(R) ) of the

semigroup of left ideals of i?,as well as subdigraphs of this graph. Graph theoretic properties of these

graphs are investigated, such as transitive closure, girth, radius, diameter, and spanning subgraphs.

Conditions on certain of these graphs are given which imply that R is regular, left duo, or that the

idempotents of R are central. We characterize simple rings in terms of Cay(IL(/?) ). We characterize

strongly regular rings in terms of a subdigraph of Cay(IL(/?)).

Key words ： Cayley graph ； semigroup ； left ideal

€L€ number： 0157.5, 0152.7, 0153.3 Document code： A

Article ID： 1000-5137(2014)05-0506-05

2010 Mathematics Subject Classification. 05C25, 05E99, 20M10.

I Introduction

A directed graph or digraph Gisa pair of sets, the set V of vertices and the set E of edges, denoted G =

(V,E) , where V 7^0 and EQV xV. Ii v ,we.V and (v,w) , then (v yw) is denoted by v —^ w. The edge

v —^ w is said to be incident from v and incident to w. An edge from v to itself is a loop. We allow multiple ed

ges between two vertices of G. Directed graphs with loops and multiple edges are often referred to as quivers. A

subdigraph of G is a digraph Gt - {Vr ,Et) ,where V' QV and Et QE.

A path of length n in G from vertex v to vertex ΐί； is a sequence of edges v v2m'm —^vn —w where

the vertices are all distinct. The distance from i; to ΐί； is the length of the shortest path from v io w. The eccen

tricity of a vertex v is the maximum of its finite distances to all other vertices. The radius of G is the minimum

eccentricity of all its vertices. The diameter of G is the maximum eccentricity of all its vertices; i. e. ,the diam

eter is the maximum distance between all pairs of distinct vertices in G.

A digraph G is strongly connected if there is a path from any vertex to any other vertex. A maximal strong

ly connected subdigraph of G is a strongly connected component of G.

A cycle from vertex v to itself in G is a path from v to some vertex w together with an edge from w to v. The

girth of G is the length of the smallest cycle in G. The circumference of G is the length of the largest cycle

in G.

If G = (V,E) is a directed graph, then the underlying graph of G is the undirected graph Gt - (V \E r ),

Received date

：

2014-07-20

Corresponding author

：

Aihua Li, Professor, E-mail

：

Iia@ mail, montcair. edu

；

Ralph P. Tucci, Professor, E-mail

：

tucci@

loyno. edu

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