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Let $T$ be a subset of a ring $A$, and let $M$ be an $A$-module. We study the additive subgroups $F$ of $M$ such that, for all $x \in M$, if $tx \in F$ for some $t \in T$, then $x \in F$. We call any such subset $F$ of $M$ a $T$-factroid of $M$, which is a kind of dual to the notion of a $T$-submodule of $M$. We connect the notion with the zero-divisors on $M$, various classes of primary and prime ideals of $A$, Euclidean domains, and the recent concepts of unit-additive commutative rings and of Egyptian fractions with respect to a multiplicative subset of a commutative ring. We also introduce a common generalization of local rings and unit-additive rings, called *sublocalizing* rings, and relate them to $T$-factroids., Comment: Removed color from erroneously colored text. Removed a nonequivalent condition from 8.13. 42 pages; comments very welcome!

Closure operations such as tight and integral closure and test ideals have appeared frequently in the study of commutative algebra. This articles serves as a survey of the authors' prior results connecting closure operations, test ideals, and interior operations via the more general structure of pair operations. Specifically, we describe a duality between closure and interior operations generalizing the duality between tight closure and its test ideal, provide methods for creating pair operations that are compatible with taking quotient modules or submodules, and describe a generalization of core and its dual. Throughout, we discuss how these ideas connect to common constructions in commutative algebra., Comment: 24 pages. Comments welcome

The *reciprocal complement* $R(D)$ of an integral domain $D$ is the subring of its fraction field generated by the reciprocals of its nonzero elements. Many properties of $R(D)$ are determined when $D$ is a polynomial ring in $n\geq 2$ variables over a field. In particular, $R(D)$ is an $n$-dimensional, local, non-Noetherian, non-integrally closed, non-factorial, atomic G-domain, with infinitely many prime ideals at each height other than $0$ and $n$., Comment: 24 pages. Comments are welcome!

A ring is *unit-additive* if a sum of units is always either a unit or nilpotent. For example, $k[X]$ and $k[X]/(X^2)$ are unit-additive, but $\mathbb Z$ is not. We prove a wide-ranging theorem about semigroup rings, showing among other things that an affine semigroup ring $A[M]$ is unit-additive if and only if $A$ is unit-additive and $M$ has no nontrivial invertible elements. In the algebraic variety setting, we get a characterization of unit additivity in terms of the nonexistence of nonconstant maps to the punctured affine line, which can be seen in terms of extensions of the Fundamental Theorem of Algebra. Specializing to elliptic curves, we show that the affine coordinate ring of an elliptic curve is always unit-additive. The concept of unit additivity leads to the related concept of unit-additivity dimension -- i.e. how far is an integral domain from being unit-additive? It turns out that rings of unit-additivity dimension 1 are of some interest, as they include the rings of integers of number fields, all power series rings, and most local rings. We construct rings of all unit-additivity dimensions and show that in the affine setting, unit-additivity dimension is bounded above by Krull dimension. We also construct the *unit-additive closure* of an integral domain $D$, being the smallest subring of the fraction field of $D$ that is unit-additive, as a localization at a certain multiplicative set in $D$., Comment: We learned that the ring property we had been calling "unit trivial'' already existed in the literature as the "UU" property; we altered the paper accordingly. We replaced the proof of Theorem 6.6 (the dimension inequality theorem) with a simpler one. Other minor grammatical changes. 24 pages; comments welcome

Let D be a Euclidean domain, with fraction field K. Let R(D) be the subring of K generated by the reciprocals of the nonzero elements of D. The main theorem states that if R(D) is not equal to K, then R(D) is a rank 1 discrete valuation ring that contains a field consisting of the units of D along with 0. Connections are made to ideas from medieval Italian mathematics., Comment: 7 pages. Comments welcome!

We systematically study the intersection flatness and Ohm-Rush properties for modules over a commutative ring, drawing inspiration from the work of Ohm and Rush and of Hochster and Jeffries. We then use our newfound understanding of these properties to deduce consequences for the Frobenius map. We establish new structural results for modules that are intersection flat/Ohm-Rush by exhibiting intimate connections between these notions and the seminal work of Raynaud and Gruson on Mittag-Leffler modules. In particular, we develop a theory of Ohm-Rush modules that is parallel to the theory of Mittag-Leffler modules. Our motivation for doing so is that the Ohm-Rush condition is weaker than intersection flatness, and it will be shown in future work that one only needs the Ohm-Rush property of Frobenius on an excellent regular ring to prove the existence of test elements for homomorphic images of the ring. We also obtain descent and local-to-global results for intersection flat/Ohm-Rush modules. Our investigations reveal a pleasing picture for flat modules over a complete local ring, in which case many otherwise distinct properties coincide. Using these results, we characterize when a regular local ring has intersection flat Frobenius in terms of a purity condition on the relative Frobenius of the completion map, we provide a new characterization of excellence for DVRs in prime characteristic, we prove new cases of intersection flatness of Frobenius, we prove descent and ascent results for intersection flatness and Ohm-Rush properties for Frobenius, and we show that proving the existence of test elements for homomorphic images of excellent regular rings reduces to exhibiting openness of pure loci for certain simple module maps. We study this openness of loci problem in a special case and show that the Frobenius of any excellent regular ring of prime characteristic is close to being Ohm-Rush., Comment: Added results on the ascent of content under relative cyclic purity assumptions (Prop. 4.4.16, Prop. 6.4.5, Rmk 6.4.6)

The notion of an \emph{Egyptian} integral domain $D$ (where every fraction can be written as a sum of unit fractions with denominators from $D$) is extended here to the notion that a ring $R$ is \emph{$W$-Egyptian}, with $W$ a multiplicative set in $R$. The new notion allows denominators just from $W$. It is shown that several results about Egyptian domains can be extended to the $W$-Egyptian context, though some cannot, as shown in counterexamples. In particular, being a sum of unit fractions from $W$ is \emph{not} equivalent to being a distinct sum of unit fractions from $W$, so we need to add the notion of the \emph{strictly} $W$-Egyptian ring. Connections are made with Jacobson radicals, power series, products of rings, factor rings, modular arithmetic, and monoid algebras., Comment: Refereed version. Expanded the introduction and added many little clarifications throughout the paper. 7 pages

The notion of an Egyptian domain (where the analogue of Egyptian fractions works appropriately), first explored by Guerrieri-Loper-Oman, is extended to the more general notions of generically and locally Egyptian domains. Results from the previous paper are extended, as well as reinterpreted in the new expanded context. It is shown that localizations of polynomial rings are typically Egyptian, that positive semigroup-graded domains are not Egyptian, and that affine semigroup rings are not Egyptian unless the semigroup is a group. However, any finitely generated algebra over a field or $\mathbb Z$ is shown to be locally Egyptian. It is further shown that if $R$ is a pullback of $S$, then $R$ is Egyptian if and only if $S$ is., Comment: Some minor errors corrected and points clarified, due to referee comments. 10 pages

In this article we first compare the set of elements in the socle of an ideal of a polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ that are not in the ideal itself and Macaulay's inverse systems of such polynomial algebras in a purely combinatorial way for monomial ideals, and then develop some closure operational properties for the related poset ${{\nats}_0^d}$. We then derive some algebraic propositions of $\Gamma$-graded rings that then have some combinatorial consequences. Interestingly, some of the results from this part that uniformly hold for polynomial rings are always false when the ring is local. We finally delve into some direct computations, w.r.t.~a given term order of the monomials, for general zero-dimensional Gorenstein ideals and deduce a few explicit observations and results for the inverse systems from some recent results about socles., Comment: 36 pages, a few new examples added, a reference added, some rephrasing of confusing sentences were made and typos fixed

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 3 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., Comment: Accepted version. 56 pages

In this note, a condition (\emph{open persistence}) is presented under which a (pre)closure operation on submodules (resp. ideals) over rings of global sections over a scheme $X$ can be extended to a (pre)closure operation on sheaves of submodules of a coherent $\mathcal{O}_X$-module (resp. sheaves of ideals in $\mathcal{O}_X$). A second condition (\emph{glueability}) is given for such an operation to behave nicely. It is shown that for an operation that satisfies both conditions, the question of whether the operation commutes with localization at single elements is equivalent to the question of whether the new operation preserves quasi-coherence. It is shown that both conditions hold for tight closure and some of its important variants, thus yielding a geometric reframing of the open question of whether tight closure localizes at single elements. A new singularity type (\emph{semi F-regularity}) arises, which sits between F-regularity and weak F-regularity. The paper ends with (1) a case where semi F-regularity and weak F-regularity coincide, and (2) a case where they cannot coincide without implying a solution to a major conjecture., Comment: According to further suggestions from the referee, I introduced a new notion, that of a *p-system* of ideals, which generalizes both p-families and F-graded systems, and I redid Section 4 to be compatible with this general notion. Accepted for publication in AMV (Acta Mathematica Vietnamica). 18 pages

We develop a duality for operations on nested pairs of modules that generalizes the duality between absolute interior operations and residual closure operations from [ER21], extending our previous results to the expanded context. We apply this duality in particular to integral and basically full closures and their respective cores to obtain integral and basically empty interiors and their respective hulls. We also dualize some of the known formulas for the core of an ideal to obtain formulas for the hull of a submodule of the injective hull of the residue field. The article concludes with illustrative examples in a numerical semigroup ring., Comment: Edited for readability and clarity. 39 pages, to appear in J. Pure Appl. Algebra

Broadening existing results in the literature to much wider classes of rings, we prove among other things: 1. Reduced quotients of excellent regular rings of characteristic $p$ admit big test elements, 2. The set of F-jumping numbers of a principal ideal in a locally excellent regular ring is a discrete subset of $\mathbb Q$, and 3. If $R$ is a quotient of a locally excellent regular ring of prime characteristic, then there is a uniform upper bound on the Hartshorne-Speiser-Lyubeznik numbers of the injective hulls of the residue fields of $R$. To do so, we develop the parallel theories of Ohm-Rush and intersection flat algebras. We show that both properties can be checked locally in flat maps of Noetherian rings. We show that intersection-flatness admits a content theory parallel to that of Ohm-Rush content for Ohm-Rush algebras. We develop descent results for these properties. Using the descent result for intersection flatness, we obtain a local condition under which a faithfully flat map of Noetherian rings must be intersection-flat. The local condition for intersection-flatness allows us to conclude that finitely generated faithfully flat algebras over a Noetherian ring are intersection-flat. Combining the local condition for intersection flatness with results of Kunz and Radu yields the conclusion that the Frobenius endomorphism associated to a locally excellent regular ring of prime characteristic is intersection-flat, thus answering a question of Sharp. Applications of the latter result include the three enumerated results above. We also get applications to tight closure and parameter test ideals., Comment: Propositions 2.6 and 4.6 are incorrect. Without them, I don't have proofs for the main theorems. I plan to incorporate many of the remaining valid statements into future joint work

An Ohm-Rush algebra $R \rightarrow S$ is called *McCoy* if for any zero-divisor $f$ in $S$, its content $c(f)$ has nonzero annihilator in $R$, because McCoy proved this when $S=R[x]$. We answer a question of Nasehpour by giving an example of a faithfully flat Ohm-Rush algebra with the McCoy property that is not a weak content algebra. However, we show that a faithfully flat Ohm-Rush algebra is a weak content algebra iff $R/I \rightarrow S/I S$ is McCoy for all radical (resp. prime) ideals $I$ of $R$. When $R$ is Noetherian (or has the more general \emph{fidel (A)} property), we show that it is equivalent that $R/I \rightarrow S/IS$ is McCoy for all ideals., Comment: 5 pages. Comments quite welcome!

One says that a ring homomorphism $R \rightarrow S$ is Ohm-Rush if extension commutes with arbitrary intersection of ideals, or equivalently if for any element $f\in S$, there is a unique smallest ideal of $R$ whose extension to $S$ contains $f$, called the content of $f$. For Noetherian local rings, we analyze whether the completion map is Ohm-Rush. We show that the answer is typically `yes' in dimension one, but `no' in higher dimension, and in any case it coincides with the content map having good algebraic properties. We then analyze the question of when the Ohm-Rush property globalizes in faithfully flat modules and algebras over a 1-dimensional Noetherian domain, culminating both in a positive result and a counterexample. Finally, we introduce a notion that we show is strictly between the Ohm-Rush property and the weak content algebra property., Comment: Little changes made throughout. 14 pages. Comments welcome!

Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely one where finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence. Connections are made with Ohm-Rush content theory, intersection-flatness of the Frobenius map, and various flatness criteria., Comment: 6 pages. Little clarifying bits added throughout. Comments welcome

Exploiting the interior-closure duality developed by Epstein and R.G., we show that for the class of Matlis dualizable modules $\mathcal{M}$ over a Noetherian local ring, when cl is a Nakayama closure and i its dual interior, there is a duality between cl-reductions and i-expansions that leads to a duality between the cl-core of modules in $\mathcal{M}$ and the i-hull of modules in $\mathcal{M}^\vee$. We further show that many algebra and module closures and interiors are Nakayama and describe a method to compute the interior of ideals using closures and colons. We use our methods to give a unified proof of the equivalence of F-rationality with F-regularity, and of F-injectivity with F-purity, in the complete Gorenstein local case. Additionally, we give a new characterization of the finitistic tight closure test ideal in terms of maps from $R^{1/p^e}$. Moreover, we show that the liftable integral spread of a module exists., Comment: 39 pages. Apart from a couple minor corrections in section 3, the main change in this version is that we spruced up the introduction. Comments still very welcome!

We define a duality operation connecting closure operations, interior operations, and test ideals, and describe how the duality acts on common constructions such as trace, torsion, tight and integral closures, and divisible submodules. This generalizes the relationship between tight closure and tight interior given in [Epstein-Schwede 2014] and allows us to extend commonly used results on tight closure test ideals to operations such as those above., Comment: 34 pages, to appear in Rocky Mountain Journal of Mathematics. Minor changes of phrasing

For a finite subset $M\subset [x_1,\ldots,x_d]$ of monomials, we describe how to constructively obtain a monomial ideal $I\subseteq R = K[x_1,\ldots,x_d]$ such that the set of monomials in $\text{Soc}(I)\setminus I$ is precisely $M$, or such that $\overline{M}\subseteq R/I$ is a $K$-basis for the the socle of $R/I$. For a given $M$ we obtain a natural class of monomials $I$ with this property. This is done by using solely the lattice structure of the monoid $[x_1,\ldots,x_d]$. We then present some duality results by using anti-isomorphisms between upsets and downsets of $(\mathbb Z^d,\preceq)$. Finally, we define and analyze zero-dimensional monomial ideals of $R$ of type $k$, where type $1$ are exactly the Artinian Gorenstein ideals, and describe the structure of such ideals that correspond to order-generic antichains in $\mathbb Z^d$., Comment: 32 pages. Minor edits. Converted to amsart format. Comments welcome

The connection between a univariate polynomial having locally principal content and the content function acting like a homomorphism (the so-called Gaussian property) has been explored by many authors. In this work, we extend several such results to the contexts of multivariate polynomials, power series over a Noetherian ring, and base change of affine $K$-algebras by separable algebraically closed field extensions. We do so by using the framework of the Ohm-Rush content function. The correspondence is particularly strong in cases where the base ring is approximately Gorenstein or the element of the target ring is regular., Comment: Removed a superfluous proposition, as a more general version already exists in the literature; added a corollary at the end and a reference to it in the Main Results block on page 2; changed the title and abstract slightly; cleaned up some things. 13 pages. Comments still welcome!

The notion of an Ohm-Rush algebra, and its associated content map, has connections with prime characteristic algebra, polynomial extensions, and the Ananyan-Hochster proof of Stillman's conjecture. As further restrictions are placed (creating the increasingly more specialized notions of weak content, semicontent, content, and Gaussian algebras), the construction becomes more powerful. Here we settle the question in the affirmative over a Noetherian ring from our previous article of whether a faithfully flat weak content algebra is semicontent (and over an Artinian ring of whether such an algebra is content), though both questions remain open in general. We show that in content algebra maps over Pr\"ufer domains, heights are preserved and a dimension formula is satisfied. We show that an inclusion of nontrivial valuation domains is a content algebra if and only if the induced map on value groups is an isomorphism, and that such a map induces a homeomorphism on prime spectra. Examples are given throughout, including results that show the subtle role played by properties of transcendental field extensions., Comment: In addition to some cosmetic improvements, the referee report allowed expansion and improvement of enough things that the old Section 2 has been split into two sections (2 and 3). In the new Section 3, it is shown that all of our properties descend to factor rings, and that most of our properties are local. To appear in J. Algebra Appl

In preparing the paper "Some extensions of Hilbert-Kunz multiplicity", we had occasion to perform an intricate set of computations pertaining to a single illustrative example. In the end, we have decided not to include the computations in the main paper, but instead to publish them here for later reference. In particular, the current preprint gives an example where the converse to one of our main theorems holds, even though it does not fit into any of the previously known cases where a converse holds., Comment: 7 pages, comments welcome. arXiv admin note: substantial text overlap with arXiv:1103.4730

We give bounds for the Hilbert-Kunz multiplicity of the product of two ideals, and we characterize the equality in terms of the tight closures of the ideals. Connections are drawn with $*$-spread and with ordinary length calculations., Comment: Since version 1, we cleaned up the exposition in the introduction and removed extraneous material from the body of the text. 9 pages, comments welcome

We further develop the notion of perinormality from our last paper, showing that it is preserved by many pullback constructions. In doing so, we introduce the concepts of relative perinormality and fragility for ring extensions., Comment: Major revision. Due to an error in the main theorem of section 3 in version 1, we have reworked the notion of fragility into three potentially separate notions. There are several related new localization results. Several other things in the paper have changed a bit, most notably Corollary 2.6 (also now corrected). 18 pages, comments welcome

We introduce a new class of integral domains, the perinormal domains, which fall strictly between Krull domains and weakly normal domains. We establish basic properties of the class, and in the case of universally catenary domains we give equivalent characterizations of perinormality. (Later on, we point out some subtleties that occur only in the non-Noetherian context.) We also introduce and explore briefly the related concept of global perinormality, including a relationship with divisor class groups. Throughout, we provide illuminating examples from algebra, geometry, and number theory., Comment: substantial changes due to comments from several people including the referee (cf. acknowledgments). Among other things, universal catenarity is necessary for the main theorem of section 4, and we have new characterizations of both perinormality and global perinormality in terms of flatness of overrings. 20 pages

The content of a polynomial over a ring $R$ is a well understood notion. Ohm and Rush generalized this concept of a content map to an arbitrary ring extension of $R$, although it can behave quite badly. We examine five properties an algebra may have with respect to this function -- content algebra, weak content algebra, semicontent algebra (our own definition), Gaussian algebra, and Ohm-Rush algebra. We show that the Gaussian, weak content, and semicontent algebra properties are all transitive. However, transitivity is unknown for the content algebra property. We then compare the Ohm-Rush notion with the more usual notion of content in the power series context. We show that many of the given properties coincide for the power series extension map over a valuation ring of finite dimension, and that they are equivalent to the value group being order-isomorphic to the integers or the reals. Along the way, we give a new characterization of Pr\"ufer domains., Comment: 14 pages. Unnecessary assumptions have been removed from Theorem 4.10 and Corollary 4.11. Minor corrections made. To appear in: Journal of Algebra and its Applications

We prove a power series ring analogue of the Dedekind-Mertens lemma. Along the way, we give limiting counterexamples, we note an application to integrality, and we correct an error in the literature., Comment: Added two references, as per referee's request. 9 pages, to appear in Proc. Amer. Math. Soc

We develop the basic properties of an essentially new closure operation on submodules, the \emph{liftable integral closure} of a submodule, including its relationships with the two prevailing notions of integral closure of submodules. We show that for a quite general class of local rings, every finite length module may be represented as a quotient of the form $T/L$, where $T$ is torsionless and integrally dependent on $L$., Comment: section 6 removed; 18 pages; comments welcome

Let $R$ be a commutative ring. It is shown that there is an order isomorphism between a popular class of finite type closure operations on the ideals of $R$ and the poset of semistar operations of finite type., Comment: The main change from the previous version is a new theorem in section 4 characterizing the standardized radical in terms of the total quotient ring. I also incorporated minor changes following the referee's comments. 12 pages, comments welcome

There are several theorems describing the intricate relationship between flatness and associated primes over commutative Noetherian rings. However, associated primes are known to act badly over non-Noetherian rings, so one needs a suitable replacement. In this paper, we show that the behavior of strong Krull primes most closely resembles that of associated primes over a Noetherian ring. We prove an analogue of a theorem of Epstein and Yao characterizing flat modules in terms of associated primes by replacing them with strong Krull primes. Also, we partly generalize a classical equational theorem regarding flat base change and associated primes in Noetherian rings. That is, when associated primes are replaced by strong Krull primes, we show containment in general and equality in many special cases. One application is of interest over any Noetherian ring of prime characteristic. We also give numerous examples to show that our results fail if other popular generalizations of associated primes are used in place of strong Krull primes., Comment: Typo corrections and minor stylistic changes. To appear in: Journal of Pure and Applied Algebra

In a paper from 2002, Bruns and Gubeladze conjectured that graded algebra retracts of polytopal algebras over a field $k$ are again polytopal algebras. Motivated by this conjecture, we prove that graded algebra retracts of Stanley-Reisner rings over a field $k$ are again Stanley-Reisner rings. Extending this result further, we give partial evidence for a conjecture saying that monomial quotients of standard graded polynomial rings over $k$ descend along graded algebra retracts., Comment: Incorporating several useful suggestions due to a referee. To appear in Journal of Pure and Applied Algebra

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an \emph{Armendariz map} between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs., Comment: Expanded first paragraph in section 6. To appear in J. Algebraic Combin. 22 pages

We introduce an operation on modules over an $F$-finite ring of characteristic $p$. We call this operation \emph{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 interior, and so in particular for the test ideal, which complement the main results of a recent paper of the second author and K. Tucker. Using our theory of phantom homology, we prove a vanishing theorem for maps of Ext. We also compare our theory to M. 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 non-normal rings, proving that the finitistic test ideal coincides with the big test ideal in some cases., Comment: References added and other minor changes. To appear in the Nagoya Mathematical Journal

Let $R$ be a reduced affine $\mathbb C$-algebra, with corresponding affine algebraic set $X$. Let $\mathcal C(X)$ be the ring of continuous (Euclidean topology) $\mathbb C$-valued functions on $X$. Brenner defined the \emph{continuous closure} $I^{\rm cont}$ of an ideal $I$ as $I\mathcal C(X) \cap R$. He also introduced an algebraic notion of \emph{axes closure} $I^{\rm ax}$ that always contains $I^{\rm cont}$, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining $f \in I^{\rm ax}$ if its image is in $IS$ for every homomorphism $R \to S$, where $S$ is a one-dimensional complete seminormal local ring. We also introduce the \emph{natural closure} $I^\natural$ of $I$. One of many characterizations is $I^\natural = I + \{f \in R: \exists n >0 \text{ with } f^n \in I^{n+1}\}$. We show that $I^\natural \subseteq I^{\rm ax}$, and that when continuous closure is defined, $I^\natural \subseteq I^{\rm cont }\subseteq I^{\rm ax}$. Under mild hypotheses on the ring, we show that $I^\natural= I^{\rm ax}$ when $I$ is primary to a maximal ideal, and that if $I$ has no embedded primes, then $I = I^\natural$ if and only if $I = I^{\rm ax}$, so that $I^{\rm cont}$ agrees as well. We deduce that in the polynomial ring $\mathbb C[x_1, \ldots, x_n]$, if $f = 0$ at all points where all of the ${\partial f \over \partial x_i}$ are 0, then $f \in ( {\partial f \over \partial x_1}, \, \ldots, \, {\partial f \over \partial x_n})R$. We characterize $I^{\rm cont}$ for monomial ideals in polynomial rings over $\mathbb C$, but we show that the inequalities $I^\natural \subset I^{\rm cont}$ and $I^{\rm cont} \subset I^{\rm ax}$ can be strict for monomial ideals even in dimension 3. Thus, $I^{\rm cont}$ and $I^{\rm ax}$ need not agree, although we prove they are equal in $\mathbb C[x_1, x_2]$., Comment: section 10 totally revamped, including a corrected error by means of a new closure operation. Many little improvements have been made throughout the paper. 48 pages, comments welcome

This article is a survey of closure operations on ideals in commutative rings, with an emphasis on structural properties and on using tools from one part of the field to analyze structures in another part. The survey is broad enough to encompass the radical, tight closure, integral closure, basically full closure, saturation with respect to a fixed ideal, and the v-operation, among others., Comment: 32 pages; minor changes to section 4; to appear in the volume "Progress in Commutative Algebra" (Sean Sather-Wagstaff, Christopher Francisco, Lee Klingler, and Janet Vassilev, eds.), de Gruyter publ

Let $R$ be an excellent Noetherian ring of prime characteristic. Consider an arbitrary nested pair of ideals (or more generally, a nested pair of submodules of a fixed finite module). We do \emph{not} assume that their quotient has finite length. In this paper, we develop various sufficient numerical criteria for when the tight closures of these ideals (or submodules) match. For some of the criteria we only prove sufficiency, while some are shown to be equivalent to the tight closures matching. We compare the various numerical measures (in some cases demonstrating that the different measures give truly different numerical results) and explore special cases where equivalence with matching tight closure can be shown. All of our measures derive ultimately from Hilbert-Kunz multiplicity., Comment: 22 pages

Let $R$ be a commutative Noetherian ring. We give criteria for flatness of $R$-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if $R$ has characteristic $p$, or more generally if it has a locally contracting endomorphism. Dualizing, we give criteria for injectivity of $R$-modules in terms of coassociated primes and (h-)divisibility of certain $\Hom$-modules. Along the way, we develop tools to achieve such a dual result. These include a careful analysis of the notions of divisibility and h-divisibility (including a localization result), a theorem on coassociated primes across a $\Hom$-module base change, and a local criterion for injectivity., Comment: 19 pages