Maximal non-pseudovaluation subrings of an integral domain

The notion of maximal non-pseudovaluation subring of an integral domain is introduced and studied. Let R ⊂ S be an extension of domains. Then R is called a maximal non-pseudovaluation subring of S if R is not a pseudovaluation subring of S, and for any ring T such that R ⊂ T ⊂ S, T is a pseudovaluation subring of S. We show that if S is not local, then there no such T exists between R and S. We also characterize maximal non-pseudovaluation subrings of a local integral domain.