摘要
设R是一个交换环,研究了R的2种图结构.首先,设N表示R的幂零根,,把R的元素作为图的顶点,2个不同的顶点x和y有一边相连接,当且仅当或者,并且x,y中至少1个不是幂零元素,,则证明了下述结果:设R是交换环,使用如上图结构,X(R)<+∞当且仅当|R|<+∞,并且此时x(R)=clique(R).其次,把R的元素看作图的顶点,2个不同顶点x和y有边相连,当且仅当Annx+Anny=R.则证明了对交换诺特环R,X(R)<+∞,并猜测x(R)=clique(R).
Let R be a commutative ring, two kinds of graph structure of R are studied.First, let N denote nilradical of R, R =R/N Considering R as a simple graph whose vertices are elements of R, such that two different elements x and y are adjacement if and ouly if or , and x≠0 or y≠0, x=xmodN. Let X(R) denote the chromatic number of the graph, thus, the following result is proved: Let R be a commtative ring, using as above graphic structure, then X(R)<+∞ if and only if |R|<+∞. Moreover X(R)=clique(R).Second, considering R as a simple graph Whose vertices are elements of R, such that twodifferent elements x and y are adjacent if and only if Annx+Anny=R, then the following result is proved: for commutative Noetherian ring R, X(R)<+∞. Mormver, X(R)=clique(R) is conjectured.
出处
《北京师范大学学报(自然科学版)》
CAS
CSCD
北大核心
1998年第3期319-323,共5页
Journal of Beijing Normal University(Natural Science)
基金
国家自然科学基金!19571009