交换环的图论性质

GRAPHIC PROPERTIES OF COMMUTATIVE RINGS

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.