交换环上的形式矩阵环的零因子和零因子图（英文）  

Zero-divisors and Zero-divisor Graphs of Formal Matrix Rings over a Commutative Ring

作　　者：唐高华 李玉 苏华东 TANG Gaohua;LI Yu;SU Huadong(School of Sciences,Beibu Gulf University,Qinzhou,Guangxi,535011,P.R.China;School of Mathematics and Statistics,Nanning Normal University,Nanning,Guangxi,530023,P.R.China;School of Mathematics and Statistics,Southwest University,Chongqing,400715,P.R.China)

机构地区：[1]北部湾大学理学院,钦州广西535011 [2]南宁师范大学数学与统计科学学院,南宁广西530023 [3]西南大学数学与统计学院,重庆400715

出　　处：《数学进展》2019年第1期99-109,共11页Advances in Mathematics（China）

基　　金：supported by NSFC(Nos.11661014,11461010,11661013);the Guangxi Science Research and Technology Development Project(No.1599005-2-13);the Guangxi Natural Science Foundation(Nos.2016GXSFDA380017,2016GXNSFCA380014);the Scientific Research Fund of Guangxi Education Department(No.KY2015ZD075)

摘　　要：本文主要研究交换环R上的形式矩阵环M_n(R;{S_(ijk)})的零因子和零因子图.首先给出了环上形式线性方程组的概念,并且得到了交换环上形式齐次线性方程组有非平凡解的充分必要条件.然后证明了A是M_n(R;{S_(ijk)})的零因子当且仅当A的行列式是R的零因子当且仅当A是R[A]的零因子.最后研究了交换环R上的形式矩阵环M_n(R;{S_(ijk)})的零因子图的性质.This paper is devoted to zero-divisors of formal matrix ring Mn(R;{sijk})over a commutative ring R.First,we introduce the notion of left(right)system of formal linear equations over a ring R、and obtain necessary and sufficient conditions for a left(right)homogeneous system of formal linear equations over a commutative ring to have a nontrivial solution.Second,we prove that an element A of Mn(R;{sijk})is a zero-divisor if and only if its determinant is a zero-divisor in J?,and if and only if>1 is a zero-divisor in R[A].Relative concepts and results on system of linear equations and matrix rings over a commutative ring are generalized.Third,we investigate properties of zero-divisor graph of the formal matrix ring Mn(R;{sijk}).

关 键 词：形式矩阵环 形式线性方程组 零因子 零因子图 

分 类 号：O153.3[理学—数学]