Math 724
Fall 2001
Final Exam
a)
Translate
the property root closed to the group of divisibility (e.g. is root closed if and
only if
is …).
b)
Translate
the property completely integrally closed to the group of divisibility
as per part a).
a)
Show
that is a partially
ordered abelian group.
b)
Construct
an integral domain with group of divisibility isomorphic to under the product
order (where
denotes the
integers).
a)
Show
that the group of divisibility of has no minimal positive elements.
b)
Construct
a valuation domain with the property that for any nonzero prime ideal there is a nonzero
prime
with
c)
Construct
a valuation domain, with the property
that the group of divisibility of
has no minimal positive elements, but every proper
localization of
(that is not a field) has a group of divisibility with
minimal positive elements.