Compressed zero-divisor graphs of matrix rings over finite fields
Đurić, Alen ; Jevđenić, Sara ; Stopar, Nik
We extend the notion of the compressed zero-divisor graph ▫$\varTheta(R)$▫ to noncommutative rings in a way that still induces a product preserving functor ▫$\varTheta$▫ from the category of finite ...unital rings to the category of directed graphs. For a finite field ▫$F$▫, we investigate the properties of ▫$\varTheta(M_n(F))$▫, the graph of the matrix ring over ▫$F$▫, and give a purely graph-theoretic characterization of this graph when ▫$n \neq 3$▫. For ▫$n \neq 2$▫ we prove that every graph automorphism of ▫$\varTheta(M_n(F))$▫ is induced by a ring automorphism of ▫$M_n(F)$▫. We also show that for finite unital rings ▫$R$▫ and ▫$S$▫, where ▫$S$▫ is semisimple and has no homomorphic image isomorphic to a field, if ▫$\varTheta(R) \cong \varTheta(S)$▫, then ▫$R \cong S$▫. In particular, this holds if ▫$S=M_n(F)$▫ with ▫$n \neq 1$▫.
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