Integral closure: Difference between revisions
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In [[ring theory]], the '''integral closure''' of a commutative unital ring ''R'' in an [[algebra over a ring|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''. | In [[ring theory]], the '''integral closure''' of a commutative unital ring ''R'' in an [[algebra over a ring|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''. | ||
Revision as of 14:42, 7 February 2009
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.