I am trying to prove that if $M$ is an $R$-module, with $R$ complete w.r.t. an ideal $\mathfrak{m}$, and $M$ is separated ($\cap_k \mathfrak{m}^k M=0$) and the images of $m_1,\dots,m_n$ generate $M/\mathfrak{m} M$, then $m_1,\dots,m_n$ generate $M$.

This appears as Exercise 7.2 in Eisenbud's Commutative Algebra text.

I am pretty stuck and would appreciate some hints.

  • 1
    $\begingroup$ This is Theorem 8.4 from Matsumura, CRT. $\endgroup$
    – user26857
    Mar 27, 2016 at 11:18
  • $\begingroup$ Thanks for the hint, by the way. I was able to figure it out from your answer below. $\endgroup$
    – user194928
    Apr 4, 2016 at 14:40

1 Answer 1


Hint. Set $N=\langle m_1,\dots,m_n\rangle$. We have $M=\mathfrak mM+N=\mathfrak m(\mathfrak mM+N)+N=\mathfrak m^2M+N$, and so on.


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