There are several ways to convert a closure or interior operation to a different operation that has particular desirable properties. In this paper, we axiomatize three ways to do so, drawing on disparate examples from the literature, including tight closure, basically full closure, and various versions of integral closure. In doing so, we explore several such desirable properties, including hereditary, residual, and cofunctorial, and see how they interact with other properties such as the finitistic property.

We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be expressed in terms of the integral closure of its row ideals, and therefore can be obtained by means of a simple computer algebra procedure. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.

We prove that the integral closure of a strongly Golod ideal in a polynomial ring over a field of characteristic zero is strongly Golod, positively answering a question of Huneke. More generally, the rational power $I_{\unicode[STIX]{x1D6FC}}$ of an arbitrary homogeneous ideal is strongly Golod for $\unicode[STIX]{x1D6FC}\geqslant 2$ and, if $I$ is strongly Golod, then $I_{\unicode[STIX]{x1D6FC}}$ is strongly Golod for $\unicode[STIX]{x1D6FC}\geqslant 1$. We also show that all the coefficient ideals of a strongly Golod ideal are strongly Golod.

Let A ⊂ B be an integral ring extension of integral domains with fields of fractions K and L, respectively. The integral degree of A ⊂ B, denoted by dA(B), is defined as the supremum of the degrees of minimal integral equations of elements of B over A. It is an invariant that lies in between dK(L) and μA(B), the minimal number of generators of the A-module B. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if A ⊂ B is simple; if A ⊂ B is projective and finite and K ⊂ L is a simple algebraic field extension; or if A is integrally closed. Furthermore, d is upper-semicontinuous if A is noetherian of dimension 1 and with finite integral closure. In general, however, d is neither sub-multiplicative nor upper-semicontinuous.

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence, we are able to prove a generalization of a result of Reid, Roberts and Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.

We introduce an operation on modules over an F-finite ring of characteristic p. We call this operation tight interior. While it exists more generally, in some cases this operation is equivalent to the Matlis dual of tight closure. Moreover, the interior of the ring itself is simply the big test ideal. We directly prove, without appeal to tight closure, results analogous to persistence, colon capturing, and working modulo minimal primes, and we begin to develop a theory dual to phantom homology. Using our dual notion of persistence, we obtain new and interesting transformation rules for tight interiors, and so in particular for the test ideal. Using our theory of phantom homology, we prove a vanishing theorem for maps of Ext. We also compare our theory with Blickle’s notion of Cartier modules, and in the process we prove new existence results for Blickle’s test submodule. Finally, we apply the theory we developed to the study of test ideals in nonnormal rings, proving that the finitistic test ideal coincides with the big test ideal in some cases.

Let A be a local Noetherian domain of Krull dimension d. Heinzer, Rotthaus, and Sally have shown that if the generic formal fiber of A has dimension d – 1, then A is birationally dominated by a 1-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of A. We explore further this correspondence between prime ideals in the generic formal fiber and 1-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

The class of $\mathfrak{m}$-full and four related classes of ideals in a local ring (R, $\mathfrak{m}$) are extended by replacing $\mathfrak{m}$ with other ideals and the resulting classes of ideals are compared. It is shown that contracted ideals are $\mathfrak{m}$-full in a local ring with infinite residue field.

In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed nonregular local rings, bounding them uniformly away from 1. Our results improve previous work of Aberbach and Enescu.

In this paper, we explore a generalization of the notion of integrality. In particular, we study a near-integrality condition that is intermediate between the concepts of integral and almost integral. This property (referred to as the $\Omega $-almost integral property) is a representative independent specialization of the standard notion of almost integrality. Some of the properties of this generalization are explored in this paper, and these properties are compared with the notion of pseudo-integrality introduced by Anderson, Houston, and Zafrullah. Additionally, it is shown that the $\Omega $-almost integral property serves to characterize the survival/lying over pairs of Dobbs and Coykendall.

Quasi-socle ideals, that is, ideals of the form I = Q: mq (q ≥ 2), with Q parameter ideals in a Buchsbaum local ring (A,m), are explored in connection to the question of when I is integral over Q and when the associated graded ring G(I) ⊕ n≥0In/In+1 of I is Buchsbaum. The assertions obtained by Wang in the Cohen-Macaulay case hold true after necessary modifications of the conditions on parameter ideals Q and integers q. Examples are explored.

Let (A,m) be a Noetherian local ring with d = dim A ≥ 2. Then, if A is a Buchsbaum ring, the first Hilbert coefficients of A for parameter ideals Q are constant and equal to where hi(A) denotes the length of the ith local cohomology module of A with respect to the maximal ideal m. This paper studies the question of whether the converse of the assertion holds true, and proves that A is a Buchsbaum ring if A is unmixed and the values are constant, which are independent of the choice of parameter ideals Q in A. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.

Let X be a smooth projective variety of dimension n in Pr, and let π:X→Pn+c be a general linear projection, with c>0. In this paper we bound the scheme-theoretic complexity of the fibers of π. In his famous work on stable mappings, Mather extended the classical results by showing that the number of distinct points in the fiber is bounded by B:=n/c+1, and that, when n is not too large, the degree of the fiber (taking the scheme structure into account) is also bounded by B. A result of Lazarsfeld shows that this fails dramatically for n≫0. We describe a new invariant of the scheme-theoretic fiber that agrees with the degree in many cases and is always bounded by B. We deduce, for example, that if we write a fiber as the disjoint union of schemes Y′ and Y′′ such that Y′ is the union of the locally complete intersection components of Y, then deg Y′+deg Y′′red≤B. Our method also gives a sharp bound on the subvariety of Pr swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ran’s ‘dimension +2 secant lemma’.

An integral domain $D$ with identity is condensed (resp., strongly condensed) if for each pair of ideals $I,\,J$ of $D,\,IJ\,=\,\{ij\,;\,i\,\in I,j\in J\,\}$ (resp., $IJ=iJ$ for some $i\,\in \,I$ or $IJ\,=Ij$ for some $j\,\in \,J$). We show that for a Noetherian domain $D,\,D$ is condensed if and only if $\text{Pic}\left( D \right)\,=0$ and $D$ is locally condensed, while a local domain is strongly condensed if and only if it has the two-generator property. An integrally closed domain $D$ is strongly condensed if and only if $D$ is a Bézout generalized Dedekind domain with at most one maximal ideal of height greater than one. We give a number of equivalencies for a local domain with finite integral closure to be strongly condensed. Finally, we show that for a field extension $k\,\subseteq K$, the domain $D=\,k+XK[[X]]$ is condensed if and only if $[K:k]\,\le \,2$ or $[K:k]\,=\,3$ and each degree-two polynomial in $k[X]$ splits over $k$, while $D$ is strongly condensed if and only if $[K:k]\,\le \,2$.

It is proved that if r* is the weak normalization of an integral domain r, then the weak normalization of the power series ring r[[x1,....xn]] is contained in R*[[X1,....Xn]]. Consequently, if R is a weakly normal integral domain, then R[[X1,....Xn]] is also weakly normal.

Let R be an integral domain with quotient field K. If R has an overling S ≠ K, such that S[X] is integrally closed, then the "algebraic degree" of K((X)) over the quotient field of R[X] is infinite. In particular, it holds for completely integrally closed domain or Noetherian domain R.