The subobject obtained by adjoining to a given subring (or subalgebra) all elements of a containing ring that satisfy a monic polynomial with coefficients in the subring; equivalently, the set of elements integral over the subring.

Principle

Adjoin precisely those elements that are roots of monic polynomials with coefficients in the base subobject; integral dependence is a finite-algebraic condition characterized by existence of such monic relations and stable under taking sums and products.

Demonstration

For the subring k[x^2,x^3] of k[x], the integral closure in k[x] is k[x] because x satisfies the monic relation t^2 - x^2 = 0 over k[x^2,x^3]; in number theory, the integral closure of Z in a number field yields its ring of integers.

Misapplication

Adjoining solutions of arbitrary polynomials (not monic) or algebraic elements without verifying integrality; confusing integral closure with algebraic closure or with radical closure leads to incorrect inclusions.

Consequence

The integral closure produces an integrally closed (normal) object in the chosen ambient ring; integrality controls finiteness and ascent/descent of properties and is central to desingularization and normalization processes.

Reversal

The dual idea is restricting to elements that are integral only in smaller extensions or taking a subring generated by non-integral elements, which loses the integrally closed property and may reintroduce singularities.

Boundary

Integral closure depends on the ambient extension: an element integral in one overring may not be integral in another. Existence and finiteness of the integral closure require hypotheses (e.g., Noetherian, finite type) for finiteness; integral closure is not functorial in arbitrary ring maps.

Semantic Tension

Integral closure competes with radical closure and algebraic closure: integral closure concerns monic polynomial relations over a base, radical closure concerns root-taking of powers, and algebraic closure concerns algebraic dependence without the monic condition—these notions intersect but are distinct.

Synthesis

Integral closure collects exactly those elements of an ambient ring that satisfy monic relations over a base subring, producing an integrally closed enlargement that plays a central role in normalization, control of singularities, and finiteness questions.