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ORDERED GROUPS An Ordered Group ''G'' is called integrally closed Iff for all elements ''a'' and ''b'' of ''G'', ''a''n ≤ ''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. COMMUTATIVE RINGS A commutative ring contained in a ring is said to be integrally closed in if every element of the Integral Closure of in is also in . Typically if one refers to a domain being integrally closed without reference to an overring, it is meant that the ring is integrally closed in it's field of fractions. If the ring is not a domain, typically being integrally closed means that every local ring is an integrally closed domain. Sometimes a domain that is integrally closed is called "normal" if it is integrally closed and being thought of as a variety. In this respect, the normalization of a Variety (or Scheme ) is simply the of the integral closure of all of the rings. REFERENCES
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