On S-valuation domains

Let D be an integral domain and let S be a multiplicative subset of D. In this paper, we study integral domains whose quotient rings are valuation domain. To do this, we introduce the concept of S-valuation domains. We define D to be an S-valuation domain if for each nonzero a, b ∈ D, there exists an element s ∈ S such that a divides sb or b divides sa. Among other things, we show that D is an S-valuation domain if and only if D_S is a valuation domain. By using this result, we give several valuation-like properties.