Integral closure

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In mathematics, in the field of commutative algebra, the concept of integral closure is a generalization of the set of all algebraic integers. It is one of many closures in mathematics.

Let S be an integral domain with R a subring of S. An element s of S is said to be integral over R if s is a root of some monic polynomial with coefficients in R. ("Monic" means that the leading coefficient is 1, the identity element of R). One can make this definition with any ring S and a subring R.

One can show that the set of all elements of S that are integral over R is a subring of S containing R; it is called the integral closure of R in S. If every element of S that is integral over R is already in R then R is said to be integrally closed in S. (So, intuitively, "integrally closed" means that R is "already big enough" to contain all the elements that are integral over R). An equivalent definition is that R is integrally closed in S if and only if the integral closure of R in S is equal to R (in general the integral closure is a superset of R). The terminology is justified by the fact that the integral closure of R in S is always integrally closed in S, and is in fact the smallest subring of S that contains R and is integrally closed in S.

In the special case where S is the field of fractions of R, the integral closure of R in S is named simply the integral closure of R, and if R is integrally closed in S, then R is said to be integrally closed. If R is reduced, then one can consider R as a subring of its total field of fractions, and one can define its integral closure of as before.

For example, the integers Z are integrally closed (the fraction field of Z is Q, and the elements of Q that are integral over Z are just the elements of Z (!), hence the integral closure of Z in Q is Z). The integral closure of Z in the complex numbers C is the set of all algebraic integers.

See also algebraic closure; this is a special case of integral closure when R and S are fields.

[edit] Integral Closure of an ideal

In commutative algebra there is also a concept of the integral closure of an ideal. The integral closure of an ideal I \subset R, usually denoted by \overline I, is the set of all elements r \in R such that there exists a monic polynomial x^n + a_{1} x^{n-1} + \ldots + a_{n-1} x^1 + a_n with a_i \in I^i with r as a root. The integral closure of an ideal is easily seen to be in the radical of this ideal.

There are alternate definitions as well.

  • r \in \overline I if there exists a c \in R not contained in any minimal prime, such that c r^n \in I^n for all sufficiently large n.
  • r \in \overline I if in the normalized blow-up of I, the pull back of r is contained in the inverse image of I. The blow-up of an ideal is an operation of schemes which replaces the given ideal with a principal ideal. The normalization of a scheme is simply the scheme corresponding to the integral closure of all of its rings.

[edit] 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.
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