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| Paper IPM / M / 8538 |
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| Abstract: | |||||
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The zero-divisor graph of a ring R is defined as the directed
graph Γ(R) that its vertices are all non-zero zero-divisors
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 zero-divisor 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(Mn(R)) ≅ Γ(Mm(S)),
then n = m, |R| = |S|, and Γ(R) ≅ Γ(S)
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