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.