Integral closure
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In abstract algebra, the concept of integral closure is a generalization of the set of all algebraic integers. It is one of many closures in mathematics.
Let S be an integral domain with R a subring of S. An element s of S is said to be integral over R if s is a root of some monic polynomial with coefficients in R. ("Monic" means that the leading coefficient is 1, the identity element of R).
One can show that the set of all elements of S that are integral over R is a subring of S containing R; it is called the integral closure of R in S. If every element of S that is integral over R is already in R then R is said to be integrally closed in S. (So, intuitively, "integrally closed" means that R is "already big enough" to contain all the elements that are integral over R). An equivalent definition is that R is integrally closed in S iff the integral closure of R in S is equal to R (in general the integral closure is a superset of R). The terminology is justified by the fact that the integral closure of R in S is always integrally closed in S, and is in fact the smallest subring of S that contains R and is integrally closed in S.
In the special case where S is the fraction field of R, the integral closure of R in S is named simply the integral closure of R, and if R is integrally closed in S, then R is said to be integrally closed.
For example, the integers Z are integrally closed (the fraction field of Z is Q, and the elements of Q that are integral over Z are just the elements of Z (!), hence the integral closure of Z in Q is Z). The integral closure of Z in the complex numbers C is the set of all algebraic integers.
See also algebraic closure; this is a special case of integral closure when R and S are fields.