Z[i]-module Question: Finding Torsion and Integer r for Module M

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SUMMARY

The discussion focuses on the structure of the \(\mathbb{Z}[i]\)-module M generated by elements \(v_1\) and \(v_2\) under the equations \((1+i)v_1+(2-i)v_2=0\) and \(3v_1+5iv_2=0\). Participants aim to determine an integer \(r \geq 0\) and a torsion \(\mathbb{Z}[i]\)-module \(T\) such that \(M \cong \mathbb{Z}[i]^r \times T\). The key task is to identify the torsion submodule and establish that \(\mathbb{Z}[i]^r\) is isomorphic to \(M/T\).

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  • Knowledge of linear combinations in module theory
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Mathematicians, algebraists, and students studying module theory, particularly those focusing on \(\mathbb{Z}[i]\)-modules and torsion elements.

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Z-module Question

Let [tex]M[/tex] be the [tex]\mathbb{Z}[/tex]-module generated by the elements [tex]v_1[/tex], [tex]v_2[/tex] such that [tex](1+i)v_1+(2-i)v_2=0[/tex] and [tex]3v_1+5iv_2=0[/tex]. Find an integer [tex]r \geq 0[/tex] and a torsion [tex]\mathbb{Z}[/tex]-module [tex]T[/tex] such that [tex]M \cong \mathbb{Z}<i>^r \times T</i>[/tex].
 
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Can you find the torsion submodule? After that, Zr should be isomorphic to M/T.
 

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