Integral extension of a ring
An extension of a commutative ring with unit element such that every element is integral over , that is, satisfies an equation of the form
where , a so-called equation of integral dependence.
An element is integral over if and only if one of the following two equivalent conditions is satisfied: 1) is an -module of finite type; or 2) there exists a faithful -module that is an -module of finite type. An integral element is algebraic over . If is a field, the converse assertion holds. Elements of the field of complex numbers that are integral over are called algebraic integers. If a ring is a module of finite type over , then every element is integral over (the converse need not be true).
Suppose that is a commutative ring, and let and be elements of that are integral over . Then and are also integral over , and the set of all elements of that are integral over forms a subring, called the integral closure of in . All rings considered below are assumed to be commutative.
If is integral over and is some -algebra, then is integral over . If is an integral extension of and is some multiplicative subset of , then the ring is integral over . An integral domain is said to be integrally closed if the integral closure of in its field of fractions is . A factorial ring is integrally closed. A ring is integrally closed if and only if for every maximal ideal the local ring is integrally closed.
Let be an integral extension of and let be a prime ideal of . Then and there exists a prime ideal of that lies above (that is, is such that ). The ideal is maximal if and only if is maximal. If is a finite extension of the field of fractions of a ring and is the integral closure of in , then there are only finitely-many prime ideals of that lie above a given prime ideal of .
Suppose that ; then is an integral extension if and only if both and are integral extensions.
|||S. Lang, "Algebra" , Addison-Wesley (1974)|
|||N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)|
Integral extension of a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_extension_of_a_ring&oldid=17415