Integrality
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In commutative algebra, the notions of an element integral over a ring, and of an integral extension of rings, are a generalization of the notions in field theory of an element being algebraic over a field, and of an algebraic extension of fields.
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[edit] Convention
The term ring will be understood to mean commuative ring (with unity).
[edit] Definiton
Let B be a ring, and A be a subring of B. An element b of B is said to be integral over A if there exists a monic polynomial f with coefficients in A such that f(b) = 0. We say that B is integral over A, or an integral extension of A, if every element of B is integral over A.
[edit] Characterization by finiteness condition
Let B be a ring, and let A be a subring of B. Given an element b in B, the following conditions are equivalent:
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- i) b is integral over A;
- ii) the subring A[b] 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.
The most commonly given proof of this theorem uses the Cayley-Hamilton theorem on determinants.
[edit] Closure properties
Using the characterization of integrality in terms of finiteness, one proves the following closure properties:
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- (Integral closure) Let A
B be rings. Then the subset C of B consisting of elements integral over A is a subring of B containing A. Thus, the sum, difference, or product of elements integral over A is also integral over A. The ring C is said to be the integral closure of A in B. - (Transitivity of integrality) Let A B C be rings, and c ∈ C. If c is integral over B and B is 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.
- (Integral closure) Let A
[edit] Integral ring homomorphisms
In the definition of integrality, the assumption that A be a subring of B can be relaxed. If f: A 
B is a ring homomorphism, that is, if B is made into an A algebra by f, then we say that f is integral, or that B is an integral A-algebra, if B is integral over the subring f(A). Previously, we had only considered the case in which f was injective. Similarly, an element of B is integral over A if it is integral over the subring f(A).
Many of the preceding considerations can be summarized in the statement that an A-algebra B is a finitely generated A-module if and only if B can be generated as an A-algebra by a finite number of elements integral over A.
