A ring R is a set equipped with two binary operations that we call multiplication and addition. We take the multiplication and addition operations to be commutative, and we assume that the ring has a unit (i.e., a multiplicative identity). We also assume that the two binary operations are compatible in a standard fashion, and satisfy certain reasonable group-theoretic requirements.

An ideal in a ring is a subset I that is closed under addition and so that, if i ∈ I and r ∈ R then ri ∈ I. Ideals come up naturally in algebraic geometry in the following way. Algebraic geometry is the study of the zero sets of polynomial functions. If p₁, ..., p_k are polynomials in the variables x₁, ..., x_n, then $${x = (x₁, ..., x_n): p₁(x) = p₂(x) = ... = p_k(x) = 0} is a variety V. The variety V has no particular algebraic structure, so it is difficult to study intrinsically. Thus a nice idea is to consider the set of all polynomials that vanish on V. This set is easily seen to be an ideal I = I(V). Let us define the radical √J of an ideal J to be the set of elements r in the ring R = k[x₁, ..., x_n] (here k is the base field) of polynomials such that rⁿ ∈ J for some positive integer n. If we let V(I) denote the variety that is the set of points at which elements at which the polynomial ideal I vanishes, then Hilbert's Nullstellensatz tells that, when we consider the ring of polynomials over an algebraically closed field, I(V(J)) = √J. This result immediately establishes the signifance and centrality of the radical.

This last idea suggests the consideration of the quotient R/Iⁿ for n an integer. Here this quotient is the set of equivalence classes under the equivalence relation r₁ ∼ r₂ if and only if r₁ - r₂ ∈ Iⁿ. It turns out that, for certain ideals and for large n, the length or dimension of R/Iⁿ is an integer-valued polynomial (the Hilbert-Samuel polynomial) in n. The coefficients of this polynomial must be rational numbers (because it is integer-valued). They are numerical invariants which carry a good deal of information about R and I. The degree of the Hilbert-Samuel polynomial equals d = dim R, and the normalized leading coefficient of the polynomial (which is d! times the leading coefficient) is an invariant that encodes much information about I and its relationship to R.

The concept of integral closure is an outgrowth of this circle of ideas. Let R be a ring and I an ideal in R. We say that an element r ∈ R is integral over I if there is a positive integer n and elements a_i ∈ Iⁱ, i = 1, ..., n, such that

rⁿ + a₁ r^n-1 + a₂ r^n-2 + ... + a_n-1r + a_n = 0 .

We call this equation the equation of integral dependence of r over I. The set of all elements of R that are integral over I is called the integral closure of I. We denote this ideal by

integral closure = I'.

In case I equals its closure then we say that I is integrally closed. A typical example of a closed ideal is a prime ideal.

Prime Ideal

If I ⊆ J are ideals then we say that J is integral over I if J ⊆ I'. Notice right away that I ⊂ I'⊂ √I. Hence the closure of an ideal is a refinement of the idea of radical.

Before we present an example let us give a piece of terminology. If R is a ring and S = {s₁, ..., s_k} are elements of R then the ideal generated by S is the set of all expressions of the form r₁ s₁ + r₂ s₂ + ... + r_k s_k for r ₁, ..., r_k ∈ R. We leave it to the reader to check that the set of such expressions actually forms an ideal. We denote the ideal by (s₁, ..., s_k).

Example:

Consider the ring R of polynomials in the two indeterminates (variables) x, y. We denote this ring by p[x,y]. Let I be the ideal generated by x² and y². Then the integral closure (x², y²)' contains the element xy. To see this, let n = 2, a₁ = 0 ∈ (x², y²), and a₂ = - x²y² ∈ (x²,y²)². Then the identity
(xy)² + a₁(xy) + a₂ = 0
is an equation of integral dependence for xy over (x², y²).

As an exercise, the reader may wish to verify that, for nonnegative integers i ≤ d, xⁱy^d-i lies in (x^d, y^d)'.

Figure 1 exhibits a pictorial method to think about the example. The dots in the plane correspond to the exponent pairs of the different monomials. What is interesting is that the fact that xy lies in the closure of (x²,y²) has the geometric interpretation that the point (1,1) (corresponding to xy) in the Cartesian plane lies in the convex hull of the set determined by (2,0) (corresponding to x²) and by (0,2) (corresponding to y²).

Figure 1

Now let us give an in-context example of how the concept of integral closure arises naturally. Let R be a ring as usual and let t be an indeterminate. Consider the ring of polynomials R[t] in the variable t with coefficients in R. Let f ∈ R[t]. The content c(f) of f is the ideal in R that is generated by the coefficients of f.

The classical Dedekind-Mertens Lemma states that if f = a₀ + a₁ t + ... + a_m t^m and g = b₀ + b₁ t + ... + b_n tⁿ are elements of R[t] then

c(f)ⁿ c(f) c(g) = c(f)ⁿ c(fg) .

We cannot provide the details of the proof of this important identity, but we note the following. Certainly the inclusion c(f) c(g) ⊇ c(fg) is obvious and elementary. It can also be shown that c(f)c(g) is integral over c(fg). This in turns leads to a sort of ``generic'' equation for integral closure.

In general it is quite difficult to actually calculate the integral closure of an ideal in a given ring. Yet there is great interest in doing so. Some workers are developing software and allied theoretical tools to carry out such calculational tasks.