Von Neumann正则环上的零因子图

讨论了一般Von Neumann正则环上的零因子图结构，重点刻画了其连通性和顶点性质.若R是有单位元的正则环，则其零因子图Γ(R)连通当且仅当R是直有限的；若R是无单位元的正则环，则其零因子图Γ(R)连通当且仅当R无真的单边恒等元；若R是满足|R|≥ 5的正则环，则其零因子图Γ(R)的源点和收点可以刻画为Sour(R)={a∈R|a是右可逆的但左不可逆}，Sink(R)={a∈R|a是左可逆的但右不可逆}.

Zero-Divisor Graph of Von Neumann Regular Rings

In this paper we investigate the zero-divisor graph of von Neumann regular rings，and we focus our main attention on the property of vertices and the connectedness of the zero-divisor graph Γ(R).Let R be a von Neumann regular ring with unit 1，we show that Γ(R) is connected if and only if R is direct finite.In addition，if R is a regular ring without unity elements，then Γ(R) is connected if and only if R has no proper one-sided identity elements.For the zero-divisor graph Γ(R) of a regular ring R with |R|≥ 5，we have that Sour(R)={a∈R|a is right invertible but not left invertible} and Sink (R)={a∈R|a is left invertible but not right invertible}.