Integral Closure of a Ring

In abstract algebra, the concept of integral closure is a generalization of the set of all algebraic integers. It is one of many closures in mathematics. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...

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). In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

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 iff 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. In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ...

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. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ...

See also algebraic closure; this is a special case of integral closure when R and S are fields. In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... This article presents the essential definitions. ...

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 , usually denoted by , is the set of all elements such that there exists a monic polynomial with with r as a root. The integral closure of an ideal is easily seen to be in the radical of this ideal. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In ring theory, a branch of mathematics, the radical of an ideal is a kind of completion of the ideal. ...

There are alternate definitions as well.

• if there exists a not contained in any minimal prime, such that for all sufficiently large n.

• 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.

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.