Some arithmetic properties on nonstandard rationals

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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.