integral extension
An integral extension is the ring-theoretic version of 'algebraic but with no fractions allowed'. Just as an algebraic number satisfies a polynomial over the rationals, an integral element satisfies a polynomial over the base ring — but with the extra demand that the leading coefficient be 1. That monic requirement is what keeps you from sneaking in denominators, so integrality is the honest analogue of being an algebraic integer rather than merely an algebraic number.
Let R be a subring of a commutative ring S. An element s of S is integral over R if it satisfies a monic polynomial s^n + c_{n-1} s^{n-1} + ... + c_0 = 0 with all c_i in R. The extension R ⊆ S is integral if every element of S is integral over R. Equivalent characterizations: s is integral over R iff R[s] is a finitely generated R-module, iff R[s] sits inside some R-subalgebra that is a finitely generated R-module.
Integrality behaves like a finiteness condition: the set of all elements of S integral over R forms a subring, the integral closure of R in S. A finite extension (S finitely generated as an R-module) is integral, but the converse fails — the algebraic closure of Q is integral over Q yet infinite-dimensional. Honest caveat: 1/2 is algebraic over Z (it satisfies 2x - 1 = 0) but NOT integral over Z, since no monic integer polynomial has 1/2 as a root; this is the whole point of the monic condition.
Z ⊆ Z[i] is integral: the Gaussian integer i satisfies the monic x^2 + 1 = 0, and a + bi satisfies x^2 - 2a x + (a^2 + b^2) = 0. So Z[i] is the ring of algebraic integers in Q(i).
The Gaussian integers are integral over Z.
Key permanence: if R ⊆ S ⊆ T with S integral over R and T integral over S, then T is integral over R (transitivity). Integral extensions preserve dimension, the foundation of going-up.