Definition 3

Let A be a ring and R a (commutative)A-algebra. The sub-A-algebra A' of R consisting of the elements of R integral over A (no. 1, Corollary 1 to Proposition 4) is called the integral closure of A in R. If A' is equal to the canonical image of A in R, A is called integrally closed in R.

Remarks

(1) If $h : A \to R$ is the ring homomorphism defining the A-algebra structure on R, the integral closure of A in R is also that of $h(A)$ in R. On the other hand, if R' is a subalgebra of R, the integral closure of A in R' is $A' \cap R'$.

(2) If A is a field, the integral closure $A'$ of A in R consists of the elements of R which are algebraic over A (no. 1, Example 1); generalizing the terminology used for field extensions (Algebra, Chapter V, § 3, no. 3), A' is then also called the algebraic closure of the field A in the algebra R and A is called algebraically closed in R if $A' = A$.