Von Neumann正则环上的零因子图

Zero-Divisor Graph of Von Neumann Regular Rings

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}.

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 graphΓ（R）.Let Rbe a von Neumann regular ring with unit 1,we show thatΓ（R）is connected if and only if Ris direct finite.In addition,if Ris a regular ring without unity elements,thenΓ（R）is connected if and only if Rhas no proper one-sided identity elements.For the zero-divisor graphΓ（R）of a regular ring R with｜R｜≥5,we have that Sour（R）={a∈R｜ais right invertible but not left invertible}and Sink（R）={a∈R｜ais left invertible but not right invertible}.