Let R be a commutative ring with 1 ≠ 0, and let Z(R) denote the set of zero-divisors of R. One can associate with R a graph Gamma( R) whose vertices are the nonzero zero-divisors of R. Two distinct vertices x and y are joined by an edge if and only if xy = 0 in R. Gamma( R) is often called the zero-divisor graph of R. We determine which finite commutative rings yield a planar zero-divisor graph. Next, we investigate the structure of Gamma(R) when Gamma( R) is an infinite planar graph. Next, it is possible to extend the definition of the zero-divisor graph to a commutative semigroup. We investigate the problem of extending the definition of the zero-divisor graph to a noncommutative semigroup, and attempt to generalize results from the commutative ring setting. Finally, we investigate the structure of Gamma(k1 x ··· x k n) where each ki is a finite field. The appendices give planar embeddings of many families of zero-divisor graphs.
展开▼
机译：令R是一个1≠0的交换环，令Z（R）表示R的零除数的集合。一个可以与R关联的图Gamma（R）的顶点是R的非零零除数。2当且仅当R中xy = 0时，不同的顶点x和y才由边连接。Gamma（R）通常被称为R的零因子图。我们确定哪些有限的交换环产生平面零因子图。接下来，当Gamma（R）是无限平面图时，我们研究Gamma（R）的结构。接下来，可以将零除数图的定义扩展到交换半群。我们研究了将零除数图的定义扩展到非交换半群的问题，并尝试归纳交换环设置的结果。最后，我们研究Gamma（k1 x···x k n）的结构，其中每个ki是一个有限域。附录给出了许多零因数图族的平面嵌入。
展开▼
