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Math Help - Ideals in Valuation Domains

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    Ideals in Valuation Domains

    Let V be a valuation integral domain. Let A be a proper ideal of V. And let B be the intersection of A, A^1, A^2,.... to infinity.

    Prove that B is a prime ideal of V

    Prove that if P is a prime ideal of V properly contained in A, then P is contained in B.
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    Quote Originally Posted by robeuler View Post
    Let V be a valuation integral domain. Let A be a proper ideal of V. And let B = \bigcap_{n=1}^{\infty}A^n.

    Prove that B is a prime ideal of V.
    suppose x,y \in V with x \notin B, \ y \notin B. we need to show that xy \notin B. so there exist integers i, j \geq 1 such that x \notin A^i, \ y \notin A^j. therefore A^i \subset <x>, \ A^j \subset <y>, because the ideals of a

    valuation ring are totally ordered. obviously A^i <y> \subseteq <x><y>=<xy>. suppose that A^i<y>=<xy>. then ry=xy, for some r \in A^i and hence x=r \in A^i, which is impossible.

    thus A^i<y> \subset <xy> and so A^{i+j} \subseteq A^i<y> \subset <xy>. hence xy \notin A^{i+j} and therefore xy \notin B. \ \Box



    Prove that if P is a prime ideal of V properly contained in A, then P is contained in B.
    if A^i \subseteq P, for some integer i \geq 1, then A \subseteq P, because P is prime. but that would be impossible because we're given that P is "properly" contained in A. so we must have P \subseteq A^i, \ \forall i \geq 1. \ \Box
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