Commuting graphs of full matrix rings over finite fields
Commuting graphs of full matrix rings over finite fields
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DOI：
10.1016/j.laa.2008.01.036
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发表时间：
2008-06
影响因子：
1.1
通讯作者：
A. Abdollahi
中科院分区：
文献类型：
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作者：
A. Abdollahi
设 R 为非交换环，Z(R) 为其中心。 R 的交换图定义为图 Г(R)，其顶点集为 R⧹Z(R)，并且两个不同的顶点在交换时通过边连接。设F是有限域，n⩾2是任意整数，R是具有恒等式的环，使得 Γ(R)≅Γ(Mn(F))，其中Mn(F)是F上n×n矩阵的环。这里我们证明|R|=|Mn(F)|。我们还证明如果 |F|是质数且n=2，则R≅M2(F)。
Let R be a non-commutative ring and Z(R) be its center. The commuting graph of R is defined to be the graph Γ(R) whose vertex set is R⧹Z(R) and two distinct vertices are joint by an edge whenever they commute. Let F be a finite field, n⩾2 an arbitrary integer and R be a ring with identity such that Γ(R)≅Γ(Mn(F)), where Mn(F) is the ring of n×n matrices over F. Here we prove that |R|=|Mn(F)|. We also show that if |F| is prime and n=2, then R≅M2(F).