Von Neumann正则环上的零因子图
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摘要
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}.
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}.
引文
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[6]ANDERSON D F,LEVY R,SHAPIRO J.Zero-Divisor Graphs,Von Neumann Regular Rings,and Boolean Algebras[J].Journal Pure and Appl.Algebra,2003,180:221-241.
[7]LU Dancheng,WU Tongsuo.The Zero-Divisor Graphs of Abelian Regular Rings[J].Northeast Math.Journal,2004,20(3):339-348.
[8]GOODEARL K R.Von Neumann Regular Rings[M].London:Pitman Publishing Limited,1979.
[2]ANDERSON D F,LIVINGSTON P S.The Zero-Divisor Graph of a Commutative Ring[J].Journal Algebra,1999,217:434-447.
[3]REDMOND S P.The Zero-Divisor Graph of a Non-Commutative Ring[J].Internat.Journal Commutative Rings,2002,1(4):203-211.
[4]WU T S.On Directed Zero-Divisor Graphs of Finite Rings[J].Disc Math.,2005,296:73-86.
[5]LEVY R,SHAPIRO J.The Zero-Divisor Graph of Von Neumann Regular Rings[J].Comm.Algebra,2002,30(2):745-750.
[6]ANDERSON D F,LEVY R,SHAPIRO J.Zero-Divisor Graphs,Von Neumann Regular Rings,and Boolean Algebras[J].Journal Pure and Appl.Algebra,2003,180:221-241.
[7]LU Dancheng,WU Tongsuo.The Zero-Divisor Graphs of Abelian Regular Rings[J].Northeast Math.Journal,2004,20(3):339-348.
[8]GOODEARL K R.Von Neumann Regular Rings[M].London:Pitman Publishing Limited,1979.