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Paper IPM / M / 7838  


Abstract:  
In a manner analogous to the commutative case, the
zerodivisor graph of a noncommutative ring R can be defined as
the directed graph \mathnormalΓ(R) that its vertices are
all nonzero zerodivisors of R in which for any two distinct
vertices x and y, x→ y is an edge if and only if
xy=0. We investigate the interplay between the ringtheoretic
properties of R and the graphtheoretic properties of
\mathnormalΓ(R). In this paper it is shown that, with
finitely many exceptions, if R is a ring and S is a finite
semisimple ring which is not a field and
\mathnormalΓ(R) ≅ \mathnormalΓ (S), then
R ≅ S. For any finite field F and each integer n\geqslant2, we prove that if R is a ring and \mathnormalΓ(R) ≅ \mathnormalΓ(M_{n}(F)), then R ≅ M_{n}(F). Redmond defined
the simple undirected graph ―\mathnormalΓ(R)
obtaining by deleting all directions on the edges in
\mathnormalΓ(R). We classify all ring R whose
―\mathnormalΓ(R) is a complete graph, a bipartite
graph or a tree.
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