Let $A$ be a local ring with maximal ideal $\mathfrak{m}$, and let $A^{\wedge}$ be the $\mathfrak{m}$-adic completion of $A$. Then $A^{\wedge}$ is a local ring with maximal ideal $\mathfrak{m}^{\wedge}=\ker(A^{\wedge}\to A/\mathfrak{m})$. In the case of the $p$-adic numbers, we have $$ \mathfrak{m}^{\wedge}=\mathfrak{m}A^{\wedge}. $$ Does this equality always hold?

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    $\begingroup$ Are you sure you typed that equation right? $\endgroup$ – Sam Cassidy Apr 28 '18 at 11:35
  • $\begingroup$ @SamCassidy Thanks, fixed. $\endgroup$ – user501746 Apr 28 '18 at 12:03
  • $\begingroup$ Anyway it seems like your question is answered on page 109 of Atiyah-Macdonald. It holds if A is Noetherian. $\endgroup$ – Sam Cassidy Apr 28 '18 at 15:15

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