Maximal non-valuative domains

The notion of maximal non-valuative domain is introduced and characterized. An integral domain R is called a maximal non-valuative domain if R is not a valuative domain but every proper overring of R is a valuative domain. Maximal non-valuative domains have at most four maximal ideals. Various properties of maximal non-valuative domains are discussed. We characterize integrally closed maximal non-valuative domains in terms of B \(\acute{\text {e}}\) $\acute{\text{e}}$ zout domains. A local non-integrally closed maximal non-valuative domain is also characterized. Conditions are given under which pseudo-valuation domains and maximal non-pseudo-valuation domains are maximal non-valuative domains.