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Math Help - Annihilators and torsion R-Modules

  1. #1
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    Annihilators and torsion R-Modules

    Let R be a PID, and let B be a torsion R-module, and let p be a prime in R.

    Prove that if pb=0 for b in B, then Ann(B) is in (p), the ideal generated by p.
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  2. #2
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    Quote Originally Posted by robeuler View Post
    Let R be a PID, and let B be a torsion R-module, and let p be a prime in R.

    Prove that if pb=0 for b in B, then Ann(B) is in (p), the ideal generated by p.
    The Ann(B) is defined as

    \text{Ann(B)} = \{ r \in R | rb = 0, \forall b \in B\},

    (T(B)=B since B is a torsion R-module, where T(B) is a torsion submodule of B).

    We see that Ann(B) is an ideal of R.

    Since R is a PID, every non-zero prime ideal is a maximal ideal. Every ideal in R (except R itself) is contained in a maximal ideal. Thus, Ann(B) is contained in a maximal ideal.

    Since pb=0 for b in B, we have p \in Ann(B) by the definition of an annhilator. We also have p \in (p), where p is a prime. Since (p) is a maximal ideal, any ideal I(except R itself) containing an element of (p) should be contained in (p). Thus, Ann(B) is in (p).
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