Some arithmetic properties on nonstandard rationals

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 $^*\BZ$-module, where $^*\BZ$ is an ultrapower of $\BZ$, and we show that Nonstandard Mordell-Weil property is equivalent to weak Mordell-Weil property in $^*K$. Next we focus on priems and prime ideals of nonstandard raional number fields. We give an infinite factorization theorem on $^*\BQ$ using valuations induced from primes of $^*\BZ$, and we classify maximal and prime ideal of $^*\BZ$ in terms of maximal filter on the set of primes of $^*\BZ$ and ordered semigroups of the valuation semigroup induced from maximal ideals of $^*\BZ$.