Unmixed local rings, symbolic topology of ideals and quasi-factorial domains

It is proved that the adic and the symbolic topologies of an ideal I of a Noetherian ring are equivalent (resp. linearly equivalent), i.e., I is a t-ideal (resp. I is an s-ideal) if and only if every quintessential (resp. essential) prime of I is a minimal prime over I. A relationship of s-ideals with their graded rings is exhibited by generalizing a theorem of Huneke. Locally unmixed rings are characterized as those Noetherian rings in which every ideal of the principal class is an s-ideal or a t-ideal. Excellent rings in which all prime ideals are s-ideals or t-ideals are studied. A relationship of a conjecture of Kaplansky with the question of existence of Noetherian rings of dimension $\geq$ 3 in which all prime ideals are s-ideals, is exhibited. This motivates a study of rings (which are called quasi-factorial domains) in which every height one prime is equimultiple. Sufficient conditions are given in order that every birational integral extension of a quasi-factorial domain is a quasi-factorial domain