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Paper   IPM / M / 8919
School of Mathematics
  Title:   Strong zero-divisor graphs of non-commutative rings
  Author(s):  M. Behboodi (Joint with R. Beyranvand)
  Status:   Published
  Journal: International Journal of Algebra
  Vol.:  2
  Year:  2008
  Pages:   25-44
  Supported by:  IPM
An element a in a ring R is called a strong zero-divisor if, either 〈a 〉〈b 〉 = 0 or 〈b 〉〈a 〉 = 0, for some 0 ≠ bR (〈x 〉 is the ideal generated by xR). Let S(R) denote the set of all strong zero-divisors of R. This notion of strong zero-divisor has been extensively studied by these authors in [8]. In this paper, for any ring R, we associate an undirected graph ~Γ(R) with vertices S(R)*=S(R)\{0}, where distinct vertices a and b are adjacent if and only if either 〈a 〉〈b 〉=0 or 〈b 〉〈a 〉=0. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of ~Γ(R). It is shown that for every ring R, every two vertices in ~Γ(R) are connected by a path of length at most 3, and if ~Γ(R) contains a cycle, then the length of the shortest cycle in ~Γ(R), is at most 4. Also we characterize all rings R whose ~Γ(R) is a complete graph or a star graph. Also, the interplay of between the ring-theoretic properties of a ring R and the graph-theoretic properties of ~Γ(Mn(R)), are fully investigated.

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