In this article, we prove that a complete Noetherian local domain of mixed characteristic $p>0$ with perfect residue field has an integral extension that is an integrally closed, almost Cohen–Macaulay domain such that the Frobenius map is surjective modulo p. This result is seen as a mixed characteristic analog of the fact that the perfect closure of a complete local domain in positive characteristic is almost Cohen–Macaulay. To this aim, we carry out a detailed study of decompletion of perfectoid rings and establish the Witt-perfect (decompleted) version of André’s perfectoid Abhyankar’s lemma and Riemann’s extension theorem.

In this paper, we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two-dimensional normal excellent local ring R, and that the Hilbert polynomial of the fiber cone of a divisorial filtration on R has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of R. The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations.

We undertake a systematic study of Lipschitz normally embedded normal complex surface germs. We prove, in particular, that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz normally embedded surface singularity.

We use quasi-orders to describe the structure of C-groups. We do this by associating a quasi-order to each compatible C-relation of a group, and then give the structure of such quasi-ordered groups. We also reformulate in terms of quasi-orders some results concerning C-minimal groups given in [5].

A complete characterization of valuation rings closed for a holomorphic derivation is given, following an idea of Seidenberg, in dimension 2.

This paper studies the Ratliff–Rush closure of ideals in integral domains. By definition, the Ratliff–Rush closure of an ideal I of a domain R is the ideal given by Ĩ := ∪(In+1 :R In), and an ideal I is said to be a Ratliff–Rush ideal if Ĩ = I. We completely characterise integrally closed domains in which every ideal is a Ratliff–Rush ideal, and we give a complete description of the Ratliff–Rush closure of an ideal in a valuation domain.

An ideal $I$ of a ring $R$ is called a radical ideal if $I\,=\,\mathcal{R}(R)$ where $\mathcal{R}$ is a radical in the sense of Kurosh–Amitsur. The main theorem of this paper asserts that if $R$ is a valuation domain, then a proper ideal $I$ of $R$ is a radical ideal if and only if $I$ is a distinguished ideal of $R$ (the latter property means that if $J$ and $K$ are ideals of $R$ such that $J\,\subset \,I\,\subset \,K$ then we cannot have $I/J\,\cong \,K/I$ as rings) and that such an ideal is necessarily prime. Examples are exhibited which show that, unlike prime ideals, distinguished ideals are not characterizable in terms of a property of the underlying value group of the valuation domain.

Let ℜ be the class of commutative rings R with comparable regular elements, that is, given two non zero-divisors in R, one divides the other. Applying the notion of V-valuation due to Harrison and Vitulli, we define the class V-val of V-valuated rings, which is contained in ℜ and contains the class of Manis valuation rings. We prove that these inclusions of classes are both proper. We investigate Prüfer rings inside ℜ, showing that there exist Prüfer rings which lie in ℜ but not in V-val; we prove that a ring R is a Prüfer valuation ring if and only if it is Prüfer and V-valuated, if and only if its lattice of regular ideals is a chain. Finally, we introduce and investigate the ideal I∞ of a ring R ∈ ℜ, which corresponds to the counterimage of ∞, whenever R is V-valuated.

A famous theorem of Krull's is that the integral closure of an integral domain D is the intersection of the valuation domains that contain D. An example is given to show that the same result need not hold for the integral closure of a ring with zero divisors.

A characterization of torsionless rings is given which shows that the integral closure of a torsionless ring need not be Prüfer.

Anderson and Ohm have introduced valuations of monoid rings k[Γ] where k is a field and Γ a cancellative torsion-free commutative monoid. We study the residue class fields in question and solve a problem concerning the pure transcendence of the residue fields.

Throughout this paper rings are understood to be commutative with unity. In this paper we prove the general approximation theorem for valuations whose infinite ideals have large Jacobson radicals. We give an example in which it is shown that approximation theorems for Manis valuations do not hold in the general case. Also we prove that every valuation pair (Rv, Pv) of a total quotient ring T(R) whose infinite ideal has large Jacobson radical is a Prüfer valuation pair.