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''. | ||
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''. | 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== | ==References== | ||
* {{cite book | author=Pierre Samuel | authorlink=Pierre Samuel | title=Algebraic number theory | publisher=Hermann/Kershaw | year=1972 }} | * {{cite book | author=Pierre Samuel | authorlink=Pierre Samuel | title=Algebraic number theory | publisher=Hermann/Kershaw | year=1972 }} | ||
* {{cite book | author=Irena Swanson | coauthors=Craig Huneke | title=Integral Closure of Ideals, Rings, and Modules | series=[[London Mathematical Society]] Lecture Notes | number=336 | isbn=0-521-68860-4 | doi=10.2277/0521688604 }} | |||
* {{cite book | title=Integral Closure: Rees Algebras, Multiplicities, Algorithms | author=Wolmer V. Vasconcelos | publisher=[[Springer-Verlag]] | year=2005 | isbn=3-540-25540-0 }}[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 16:00, 1 September 2024
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.