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Paper IPM / M / 8919  


Abstract:  
An element a in a ring R is called a strong
zerodivisor if, either 〈a 〉〈b 〉 = 0 or 〈b 〉〈a 〉 = 0,
for some 0 ≠ b ∈ R (〈x 〉 is the ideal generated by x ∈ R).
Let S(R) denote the set of all strong zerodivisors of R. This notion of
strong zerodivisor 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 ringtheoretic properties of R and the graphtheoretic
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 ringtheoretic properties of a ring
R and the graphtheoretic properties of ~Γ(M_{n}(R)), are fully investigated.
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