  For a given number field $K$, we show that the ranks of nonsingular elliptic
curves over $K$ are uniformly finitely bounded if and only if weak Mordell-Weil
property holds in all(some) ultrpowers $^*K$ of $K$. Also we introduce
Nonstandard Mordell-Weil property for $^*K$ considering each Mordell-Weil group
as $^*Z$-module, where $^*Z$ is an ultrapower of $Z$, and we show that
Nonstandard Mordell-Weil property is equivalent to weak Mordell-Weil property
in $^*K$. In Appendix, we showed that it is possible to consider definable
abelian groups as $^*Z$-modules in a saturated nonstandard rational number
field $^*Q$ so that nonstandard Mordell-Weil property is well-defined, and thus
we showed that nonstandard Mordell-Weil property and weak Mordell-Weil property
are equivalent.
  Next we focus on priems and prime ideals of nonstandard raional number
fields. We give an infinite factorization theorem on $^*Q$ using valuations
induced from primes of $^*Z$, and we classify maximal and prime ideal of $^*Z$
in terms of maximal filter on the set of primes of $^*Z$ and ordered semigroups
of the valuation semigroup induced from maximal ideals of $^*Z$.
