All rings are assumed to be commutative with unit and noetherian. Let \({\mathbb{P}}\) be a property which makes sense for noetherian local rings, e.g. regular, normal, reduced, Cohen-Macaulay, \((S_ n)\), \((R_ n)\) etc. For properties \({\mathbb{P}}\) verifying some list of axioms, and which do not verify all usual axioms (e.g. \((R_ n)\), \((T_ n)\) etc.), the author gives answers, under very general and natural assumptions, to a problem of Rotthaus about lifting of openness of \({\mathbb{P}}\)-loci and to a question of Valabrega about a useful criterion for \({\mathbb{P}}\)-morphisms.