In mathematics, in the area of commutative algebra, **tight closure** is an operation defined on ideals in positive characteristic. It was introduced by MelvinHochster and Craig Huneke ( 1988 , 1990 ).

Let be a commutative noetherian ring containing a field of characteristic . Hence is a prime number.

Let be an ideal of . The tight closure of , denoted by , is another ideal of containing . The ideal is defined as follows.

- if and only if there exists a , where is not contained in any minimal prime ideal of , such that for all . If is reduced, then one can instead consider all .

Here is used to denote the ideal of generated by the 'th powers of elements of , called the th Frobenius power of .

An ideal is called tightly closed if . A ring in which all ideals are tightly closed is called weakly -regular (for Frobenius regular). A previous major open question in tight closure is whether the operation of tight closure commutes with localization, and so there is the additional notion of -regular, which says that all ideals of the ring are still tightly closed in localizations of the ring.

Brenner & Monsky (2010) found a counterexample to the localization property of tight closure. However, there is still an open question of whether every weakly -regular ring is -regular. That is, if every ideal in a ring is tightly closed, is it true that every ideal in every localization of that ring is also tightly closed?

In mathematics, specifically abstract algebra, an **integral domain** is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element *a* has the cancellation property, that is, if *a* ≠ 0, an equality *ab* = *ac* implies *b* = *c*.

In mathematics, a **ring** is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

In commutative algebra, the **Krull dimension** of a commutative ring *R*, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

In mathematics, a **unique factorization domain** (**UFD**) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain in which every non-zero non-unit element can be written as a product of prime elements, uniquely up to order and units.

In ring theory, a branch of abstract algebra, a **commutative ring** is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In commutative algebra and algebraic geometry, **localization** is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing one so that it consists of fractions such that the denominator *s* belongs to a given subset *S* of *R*. If *S* is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the ring **Q** of rational numbers from the ring **Z** of rational integers.

In mathematics, a **Cohen–Macaulay ring** is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.

In commutative algebra, a **Gorenstein local ring** is a commutative Noetherian local ring *R* with finite injective dimension as an *R*-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense.

In mathematics, a **sober space** is a topological space *X* such that every irreducible closed subset of *X* is the closure of exactly one point of *X*: that is, this closed subset has a unique generic point.

In mathematics, **ideal theory** is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra. The name grew out of the central considerations, such as the Lasker–Noether theorem in algebraic geometry, and the ideal class group in algebraic number theory, of the commutative algebra of the first quarter of the twentieth century. It was used in the influential van der Waerden text on abstract algebra from around 1930.

**Melvin Hochster** is an American mathematician working in commutative algebra. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan.

In commutative algebra, an element *b* of a commutative ring *B* is said to be **integral over***A*, a subring of *B*, if there are *n* ≥ 1 and *a*_{j} in *A* such that

In commutative algebra, a **Krull ring** or **Krull domain** is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull (1931). They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1.

In commutative algebra, an **integrally closed domain***A* is an integral domain whose integral closure in its field of fractions is *A* itself. Spelled out, this means that if *x* is an element of the field of fractions of *A* which is a root of a monic polynomial with coefficients in *A,* then *x* is itself an element of *A.* Many well-studied domains are integrally closed: fields, the ring of integers **Z**, unique factorization domains and regular local rings are all integrally closed.

In algebra, the **integral closure** of an ideal *I* of a commutative ring *R*, denoted by , is the set of all elements *r* in *R* that are integral over *I*: there exist such that

**Craig Lee Huneke** is an American mathematician specializing in commutative algebra. He is a professor at the University of Virginia.

**Karen Ellen Smith** is an American mathematician, specializing in commutative algebra and algebraic geometry. She completed her bachelor's degree in mathematics at Princeton University before earning her PhD in mathematics at the University of Michigan in 1993. Currently she is the Keeler Professor of Mathematics at the University of Michigan. In addition to being a researcher in algebraic geometry and commutative algebra, Smith with others wrote the textbook *An Invitation to Algebraic Geometry*.

In algebra, the **Hilbert–Kunz function** of a local ring of prime characteristic *p* is the function

In algebra and algebraic geometry, given a commutative Noetherian ring and an ideal in it, the ** n-th symbolic power** of is the ideal

- Brenner, Holger; Monsky, Paul (2010), "Tight closure does not commute with localization",
*Annals of Mathematics*, Second Series,**171**(1): 571–588, arXiv: 0710.2913 , doi:10.4007/annals.2010.171.571, ISSN 0003-486X, MR 2630050 - Hochster, Melvin; Huneke, Craig (1988), "Tightly closed ideals",
*American Mathematical Society. Bulletin. New Series*,**18**(1): 45–48, doi: 10.1090/S0273-0979-1988-15592-9 , ISSN 0002-9904, MR 0919658 - Hochster, Melvin; Huneke, Craig (1990), "Tight closure, invariant theory, and the Briançon–Skoda theorem",
*Journal of the American Mathematical Society*,**3**(1): 31–116, doi:10.2307/1990984, ISSN 0894-0347, JSTOR 1990984, MR 1017784

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