Integrally closed

Integrally closed

In mathematics, more specifically in abstract algebra, the concept of integrally closed has two meanings, one for groups and one for rings.

Commutative rings

Main article: Integrally closed domain

A commutative ring R contained in a ring S is said to be integrally closed in S if R is equal to the integral closure of R in S. That is, for every monic polynomial f with coefficients in R, every root of f belonging to S also belongs to R. Typically if one refers to a domain being integrally closed without reference to an overring, it is meant that the ring is integrally closed in its field of fractions, the largest overring of the domain.

If the ring is not a domain, typically being integrally closed means that every local ring is an integrally closed domain.

Sometimes a domain that is integrally closed is called "normal" if it is integrally closed and being thought of as a variety. In this respect, the normalization of a variety (or scheme) is simply the Spec of the integral closure of all of the rings.

Ordered groups

An ordered group G is called integrally closed if and only if for all elements a and b of G, if an ≤ b for all natural n then a ≤ 1.

This property is somewhat stronger than the fact that an ordered group is Archimedean. Though for a lattice-ordered group to be integrally closed and to be Archimedean is equivalent. We have the surprising theorem that every integrally closed directed group is already abelian. This has to do with the fact that a directed group is embeddable into a complete lattice-ordered group if and only if it is integrally closed. Furthermore, every archimedean lattice-ordered group is abelian.

References

  • R. Hartshorne, Algebraic Geometry, Springer-Verlag (1977)
  • M. Atiyah, I. Macdonald Introduction to commutative algebra Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969
  • H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.
  • A.M.W Glass, Partially Ordered Groups, World Scientific, 1999

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Integrally closed domain — In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in the field of fractions of A is A itself. Many well studied domains are integrally closed: Fields, the ring of integers Z, unique factorization… …   Wikipedia

  • Integrality — In commutative algebra, the notions of an element integral over a ring (also called an algebraic integer over the ring), and of an integral extension of rings, are a generalization of the notions in field theory of an element being algebraic over …   Wikipedia

  • Integral element — In commutative algebra, an element b of a commutative ring B is said to be integral over its subring A if there are such that That is to say, b is a root of a monic polynomial over A.[1] If B consists of elements that are integral over A, then B… …   Wikipedia

  • Dedekind domain — In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily …   Wikipedia

  • Valuation ring — In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F , at least one of x or x 1 belongs to D .Given a field F , if D is a subring of F such that either x or x 1 belongs to D for… …   Wikipedia

  • Commutative ring — 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. Some specific kinds of commutative rings are given with …   Wikipedia

  • Glossary of scheme theory — This is a glossary of scheme theory. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of… …   Wikipedia

  • Outline of algebraic structures — In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety. Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither… …   Wikipedia

  • Field (mathematics) — This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambiguation). In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it …   Wikipedia

  • Emmy Noether — Amalie Emmy Noether Born 23 March 1882(1882 03 23) …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”