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Encyclopedia > Integrally closed

An ordered group G is called integrally closed iff for all elements a and b of G, an ≤ b for arbitrary high natural n implies a ≤ 1.


This property is somewhat stronger than that an ordered group is archimedean. Though for a lattice-ordered group to be integrally closed and to be archimedean is equivalent. We have the surprising theorem that every integrally closed directed group is already abelian. This have to do with the fact that an directed group is embeddable into a complete lattice-ordered group iff it is integrally closed. Further every archimedean lattice-ordered group is abelian.


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INTEGRALLY CLOSED (355 words)
An equivalent definiton 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''.
The integral closure of Z in the complex numbers C is the set of all algebraic integers.
Integral closure - Wikipedia, the free encyclopedia (381 words)
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).
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.
The integral closure of Z in the complex numbers C is the set of all algebraic integers.
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