# Integral closure

Revision as of 14:42, 7 February 2009 by imported>Bruce M. Tindall

In ring theory, the **integral closure** of a commutative unital ring *R* in an algebra *S* over *R* is the subset of *S* consisting of all elements of *S* integral over *R*: that is, all elements of *S* satisfying a monic polynomial with coefficients in *R*. The integral closure is a subring of *S*.

An example of integral closure is the ring of integers or maximal order in an algebraic number field *K*, which may be defined as the integral closure of **Z** in *K*.

The **normalisation** of a ring *R* is the integral closure of *R* in its field of fractions.

## References

- Pierre Samuel (1972).
*Algebraic number theory*. Hermann/Kershaw. - Irena Swanson; Craig Huneke.
*Integral Closure of Ideals, Rings, and Modules*. DOI:10.2277/0521688604. ISBN 0-521-68860-4. - Wolmer V. Vasconcelos (2005).
*Integral Closure: Rees Algebras, Multiplicities, Algorithms*. Springer-Verlag. ISBN 3-540-25540-0.