An element b of a ring extension R ⊆ S is integral over R if it satisfies a monic polynomial with coefficients in R. When every element of S is integral over R, we call R ⊆ S an integral extension. This concept generalizes algebraic field extensions to the ring setting and governs how prime ideals in R relate to prime ideals in S — the foundation for the going-up and going-down theorems.
In field theory, you studied algebraic extensions: a field extension K ⊆ L where every element of L satisfies a polynomial with coefficients in K. Integral extensions adapt this concept to rings, with one critical change — the polynomial must be monic (leading coefficient 1). An element b in a ring extension R ⊆ S is integral over R if there exist a₀, ..., aₙ₋₁ ∈ R such that bⁿ + aₙ₋₁bⁿ⁻¹ + ··· + a₁b + a₀ = 0. The insistence on monic is essential: 1/2 satisfies 2x - 1 = 0 over ℤ but is not integral over ℤ, because no monic integer polynomial vanishes at 1/2.
There is a powerful equivalent characterization: b is integral over R if and only if R[b] is a finitely generated R-module. For an algebraic element over a field, the analogous statement (K(α) is finite-dimensional over K) is familiar. But over rings, module-finiteness is strictly stronger than ring-finiteness. The element 1/2 generates ℤ[1/2] = {a/2ⁿ : a ∈ ℤ, n ≥ 0}, which is not a finitely generated ℤ-module — it requires arbitrarily large powers of 2 in the denominator. The module-finiteness criterion is the workhorse for proving transitivity of integral dependence and showing that the set of integral elements forms a ring.
The integral closure of R in S is the set of all elements of S that are integral over R; it is a subring of S containing R. A domain R is integrally closed (or normal) if its integral closure in its own fraction field is just R itself. All UFDs are integrally closed (in particular, ℤ, polynomial rings over fields, and PIDs). The ring ℤ[√-3] is the standard example of a domain that is not integrally closed: (1 + √-3)/2 is in the fraction field and is integral over ℤ (satisfying x² - x + 1 = 0), but does not lie in ℤ[√-3].
Integral extensions control the geometry of the map Spec(S) → Spec(R). The lying-over theorem says every prime of R is the contraction of some prime of S. The incomparability theorem says distinct primes of S lying over the same prime of R are not comparable by inclusion. Combined with the going-up theorem, these results show that integral extensions preserve Krull dimension and provide the algebraic backbone for studying finite morphisms in algebraic geometry. The theory also undergirds algebraic number theory: the ring of integers in a number field is the integral closure of ℤ in that field, and its structure (Dedekind domain, class group, ramification) is the central object of study.