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

[mathsss2]

Z-module Question

Let M be the \mathbb{Z}-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}-module T such that M \cong \mathbb{Z}<i>^r \times T</i>.

 

[adriank]

Can you find the torsion submodule? After that, Z^r should be isomorphic to M/T.