On directed zero-divisor graphs of finite rings
On directed zero-divisor graphs of finite rings
复制标题
DOI：
10.1016/j.disc.2005.03.006
复制
发表时间：
2004-03
期刊：
影响因子：
--
通讯作者：
Tongsuo Wu
中科院分区：
文献类型：
--
作者：
Tongsuo Wu
对于 Artinian 环 R，当且仅当 R 中不存在真单边恒等元时，有向零除数图 Γ(R) 是连通的。对于有限环 R，我们对汇和源进行了表征和澄清。特别是，证明了对于任意环 R，如果 γ(R) 中存在源 y，且 y2=0，则 |R|=4 且 R={0,x,y,z}，其中留下 x 和 z 单位元且 yx=0=yz。这样的环 R 也是唯一一个使得 Γ(R) 具有唯一源的环。这表明 Γ(R) 不能是任何有限或无限环 R 的网络。
For an artinian ring R, the directed zero-divisor graph Γ(R) is connected if and only if there is no proper one-sided identity element in R. Sinks and sources are characterized and clarified for a finite ring R. Especially, it is proved that for any ring R, if there exists a source y in Γ(R) with y2=0, then |R|=4 and R={0,x,y,z}, where x and z are left identity elements and yx=0=yz. Such a ring R is also the only ring such that Γ(R) has exactly one source. This shows that Γ(R) cannot be a network for any finite or infinite ring R.