Integral Element - Equivalent Definitions

Equivalent Definitions

Let B be a ring, and let A be a subring of B. Given an element b in B, the following conditions are equivalent:

• (i) b is integral over A;
• (ii) the subring A of B generated by A and b is a finitely generated A-module;
• (iii) there exists a subring C of B containing A and which is a finitely-generated A-module;
• (iv) there exists a finitely generated A-submodule M of B with and the annihilator of M in B is zero.

The usual proof of this uses the following variant of the Cayley–Hamilton theorem on determinants (or simply Cramer's rule.)

Theorem Let u be an endomorphism of an A-module M generated by n elements and I an ideal of A such that . Then there is a relation:

This theorem (with I = A and u multiplication by b) gives (iv) (i) and the rest is easy. Coincidentally, Nakayama's lemma is also an immediate consequence of this theorem.

It follows from the above that the set of elements of B that are integral over A forms a subring of B containing A. It is called the integral closure of A in B. The proof is due to Dedekind (Milne, ANT). If A happens to be the integral closure of A in B, then A is said to be integrally closed in B. If A is reduced (e.g., an integral domain) and B its total ring of fractions, one often drops qualification "in B" and simply says "integral closure" and "integrally closed." Let A be an integral domain with the field of fractions K and A' the integral closure of A in K. Then the field of fractions of A' is K; in particular, A' is an integrally closed domain.

Similarly, "integrality" is transitive. Let C be a ring containing B and c in C. If c is integral over B and B integral over A, then c is integral over A. In particular, if C is itself integral over B and B is integral over A, then C is also integral over A.

Note that (iii) implies that if B is integral over A, then B is a union (equivalently an inductive limit) of subrings that are finitely generated A-modules.

If A is noetherian, (iii) can be weakened to:

(iii) bis There exists a finitely generated A-submodule of B that contains .

Finally, the assumption that A be a subring of B can be modified a bit. If f: A B is a ring homomorphism, then one says f is integral if B is integral over . In the same way one says f is finite (B finitely generated A-module) or of finite type (B finitely generated A-algebra). In this viewpoint, one says that

f is finite if and only if f is integral and of finite-type.

Or more explicitly,

B is a finitely generated A-module if and only if B is generated as A-algebra by a finite number of elements integral over A.