The following fact falls under the category of Iwasawa modules.
Let $M$ be a torsion free finitely generated module over the non commutative noetherian ring $\Bbb{Z}_p[[G]]$, (where $G$ is a p analytic group) of rank $r$. I want to show that there exists $V$ free modules over $\Bbb{Z}_p[[G]]$ of rank $r$ such that $V$ is contained in $M$, with $M/V$ a torsion $\Bbb{Z}_p[[G]]$ module.
I have the following proof: Is it correct?
I found out the following proof: Is it correct? I take $r=3$. Let $A$ denote $\Bbb{Z_p[[G]]}$. Let $K=Frac(A)$ Consider $m_1\in M$. and consider the subspace $mA$. If it is the whole space $M$ then since $M$ is torsion free $dim_K(M\otimes _A K)=1$ which is a contradiction!. So there exists non zero $m_2$ not in the $A$ span of $m_1$. Now if $M=m_2A+m_1A$ then $M$ will be isomorphic to $A^2$ and so $dim_K(M\otimes _A K)=2$. Again we get contradiction. So there exists $m_3$ not in the $A$ subspace generated by $m_1$ and $m_2$. Let $P$ be $m_1A+m_2A+m_3A$. Ten $p$ is isomorphic to $A^3$ and then $A^3$ is contained in $M$. now it is easy to see that $M/P$ is torsion because both of $P$ and $M$ have same rank.
Is this proof correct?
