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!
2026-04-23 19:21:24.1776972084
a submoudule can be both essential and superfluous
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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.