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I am reading Rings and Categories of Modules by Frank W.Aderson,on 73 pages,I can't understand the statement in the picture.I can't found a submodule is both essential and superfluous.I hope someone can help me,thanks!

guojm
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1 Answers1

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Your excerpt from Anderson and Fuller consists of two sentences: the first one is the claim and the second one is a module in which every nontrivial submodule is an example. So it is hard to understand why you are asking unless you simply don't understand how to find a single submodule of $\mathbb Z_{p^\infty}$. If that's the case you should probably say something explicitly or else you look very foolish.

It is not hard to prove, or to look up, what the submodules look like. It turns out they are linearly ordered, and that is why each nontrivial submodule is both superfluous and essential.

If you need a smaller example, just use the quotient ring $F_2[X]/(X^2)$. This ring has four elements and exactly three ideals (linearly ordered) and that one nontrivial ideal is also an example.

rschwieb
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  • Thanks for your answer!You are right,I can't found a submoudule of Zp∞.In fact,I don't know what Zp∞ is.Can you help me explain it.Thank you very much! – guojm Mar 26 '17 at 08:30
  • @guojm click the link I put in the answer and you will find out – rschwieb Mar 26 '17 at 10:45
  • Thank you very much!Can you explain the quotient ring F2[X]/(X2) to me?Sorry to bother you. – guojm Mar 26 '17 at 10:53
  • @guojm You should look for posts on quotient rings here. I don't really feel like reexplaining in a comment. It's isomorphic to the matrix ring ${\begin{bmatrix}0 & 0 \ 0 & 0\end{bmatrix},\begin{bmatrix}1 & 0 \ 0 & 1\end{bmatrix},\begin{bmatrix}0 & 1 \ 0 & 0\end{bmatrix},\begin{bmatrix}1 & 1 \ 0 & 1\end{bmatrix}}$. – rschwieb Mar 26 '17 at 11:03
  • I got it.Thanks! – guojm Mar 26 '17 at 11:09
  • Sorry to bother you again.What you said above is not a ring. – guojm Mar 26 '17 at 14:19
  • @guojm Perhaps I wasn't explicit enough that the matrices have entries from the field of two elements. It is a ring. – rschwieb Mar 26 '17 at 16:02
  • OK！Thanks！I got it! – guojm Mar 27 '17 at 01:12