Let $A$ be a local noetherian ring with maximal ideal $m$. Let $M$ be an infinitely generated $A$-module and $\hat{M}$ be the $m$-adic completion of $M$. Denote by $\hat{A}$ the $m$-adic completion of $A$. Recall, $\hat{M}$ is an $\hat{A}$-module. My question is: Is $\hat{M}$ infinitely generated as an $\hat{A}$-module?

EDIT Assume that the annihilator of $M$ properly contains the zero prime ideal (the prime ideal containing $0$).

Let $A$ be a local noetherian ring with maximal ideal $m$. Let $M$ be an infinitely generated $A$-module and $\hat{M}$ be the $m$-adic completion of $M$. Denote by $\hat{A}$ the $m$-adic completion of $A$. Recall, $\hat{M}$ is an $\hat{A}$-module. My question is: Is $\hat{M}$ infinitely generated as an $\hat{A}$-module?

EDIT Assume that the annihilator of $M$ properly contains the zero prime ideal (the prime ideal containing $0$) and there does not exist an integer $n$ such that $m^n$ is contained in the annihilator of $M$.

Let $A$ be a local noetherian ring with maximal ideal $m$. Let $M$ be an infinitely generated $A$-module and $\hat{M}$ be the $m$-adic completion of $M$. Denote by $\hat{A}$ the $m$-adic completion of $A$. Recall, $\hat{M}$ is an $\hat{A}$-module. My question is: Is $\hat{M}$ infinitely generated as an $\hat{A}$-module?

Let $A$ be a local noetherian ring with maximal ideal $m$. Let $M$ be an infinitely generated $A$-module and $\hat{M}$ be the $m$-adic completion of $M$. Denote by $\hat{A}$ the $m$-adic completion of $A$. Recall, $\hat{M}$ is an $\hat{A}$-module. My question is: Is $\hat{M}$ infinitely generated as an $\hat{A}$-module?

EDIT Assume that the annihilator of $M$ properly contains the zero prime ideal (the prime ideal containing $0$).