integrality

In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n = 1 and such that

That is to say, b is a root of a monic polynomial over A. If every element of B is integral over A, then it is said that B is integral over A, or equivalently B is an integral extension of A. If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).

Olkoon B rengas, jolla on alirenkaana A. Sanotaan, että A:n kokonaissulkeuma B:ssä on osajoukko , joka sisältää kaikki ne B:n alkiot, jotka ovat kokonaisia A:ssa. Tämä tarkoittaa sitä, että xⁿ+a_n-1x^n-1 + ... + a₀ = 0, jollain a₀, a₁, ... , a_n-1 ∈ A.