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In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Γ_*(R), is a directed graph defined on its nonzero quasi-zero-divisors, where there is an arc from a vertex x to another vertex y if and only if x Ry = 0. We show that the following three conditions on an FIC ring R are equivalent:(1) χ(R) is finite;(2) ω(R) is finite;(3)Nil_*R is finite where Nil_*R equals the finite intersection of prime ideals. Furthermore, we also completely determine the connectedness, the diameter and the girth of Γ_*(R).  相似文献   
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