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

A commutative ring [math]\displaystyle{ R }[/math] contained in a ring [math]\displaystyle{ S }[/math] is said to be integrally closed in [math]\displaystyle{ S }[/math] if [math]\displaystyle{ R }[/math] is equal to the integral closure of [math]\displaystyle{ R }[/math] in [math]\displaystyle{ S }[/math]. That is, for every monic polynomial f with coefficients in [math]\displaystyle{ R }[/math], every root of f belonging to S also belongs to [math]\displaystyle{ R }[/math]. 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.

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 [math]\displaystyle{ \operatorname{Spec} }[/math] 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