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Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can talk of their dimensions. Now what can we say about rank of $M$ from looking at the dimensions of $M[p]$ and $M/(p) ?$

Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can talk of their dimensions. Now what can we say about the rank of $M$ from looking at the dimensions of $M[p]$ and $M/(p) ?$

Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}[[T]] $ vector spaces. So we can talk of their dimensions. Now what can we say about rank of $M$ from looking at the dimensions of $M[p]$ and $M/(p) ?$

Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can talk of their dimensions. Now what can we say about rank of $M$ from looking at the dimensions of $M[p]$ and $M/(p) ?$

Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p} $ vector spaces. So we can talk of their dimensions. Now what can we say about rank of $M$ from looking at the dimensions of $M[p]$ and $M/(p) ?$

Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}[[T]] $ vector spaces. So we can talk of their dimensions. Now what can we say about rank of $M$ from looking at the dimensions of $M[p]$ and $M/(p) ?$