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Math Help - Z[i]-module Question

  1. #1
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    Z[i]-module Question

    Let M be the \mathbb{Z}[i]-module generated by the elements v_1, v_2 such that (1+i)v_1+(2-i)v_2=0 and 3v_1+5iv_2=0. Find an integer r \geq 0 and a torsion \mathbb{Z}[i]-module T such that M \cong \mathbb{Z}[i]^r \times T.
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  2. #2
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    Quote Originally Posted by Erdos32212 View Post
    Let M be the \mathbb{Z}[i]-module generated by the elements v_1, v_2 such that (1+i)v_1+(2-i)v_2=0 and 3v_1+5iv_2=0. Find an integer r \geq 0 and a torsion \mathbb{Z}[i]-module T such that M \cong \mathbb{Z}[i]^r \times T.
    r=0 because M is a torsion \mathbb{Z}[i] module. this is very easy to see: 0=3(1-i)[(1+i)v_1 + (2-i)v_2]-2(3v_1+5iv_2)=(3-19i)v_2. thus v_2 is torsion. similarly you can show that v_1 is

    torsion too. thus M is a torsion module. \Box
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  3. #3
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    Thanks! I have only a few questions. How do we show that v_1 is torsion? I know how you did it for v_2, but getting things to cancel in complex analysis is not so easy [I am having trouble getting v_2 to cancel]. Also, did we find the torsion \mathbb{Z}[i]-module such that M \cong \mathbb{Z}[i]^r \times T or does T=M?.
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  4. #4
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    Quote Originally Posted by Erdos32212 View Post

    Thanks! I have only a few questions. How do we show that v_1 is torsion? I know how you did it for v_2, but getting things to cancel in complex analysis is not so easy [I am having trouble getting v_2 to cancel].
    0=(2-i)(3v_1+5iv_2) - 5i[(1+i)v_1 + (2-i)v_2]=(11-8i)v_1. so since every v \in M is a linear combination of v_1,v_2, we will have (11-8i)(3-19i)v=0.

    Also, did we find the torsion \mathbb{Z}[i]-module such that M \cong \mathbb{Z}[i]^r \times T or does T=M?.
    since M is torsion, we can just let r=0 and T=M.
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