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Paper IPM / M / 8538  


Abstract:  
The zerodivisor graph of a ring R is defined as the directed
graph Γ(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. Recently, it
has been shown that for any finite ring R, Γ(R) has an even
number of edges. Here we give a simple proof for this result. In
this paper we investigate some properties of zerodivisor graphs
of matrix rings and group rings. Among other results, we prove
that for any two finite commutative rings R,S with identity and
n,m\geqslant 2, if Γ(M_{n}(R)) ≅ Γ(M_{m}(S)),
then n = m, R = S, and Γ(R) ≅ Γ(S)
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