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Am preparing for exam few days to go. I came across this problem in Anderson - Fuller book about modules.

(1) Give an example of an indecomposable module that has a decomposable submodule.

(2) Give an example of an indecomposable module that has a decomposable factor module.

In part (1), there is a hint : Try a factor module of $R$ (as a left module over itself), where $R = Q[X,Y]$, that is the polynomial ring over rational numbers with two indeterminants. The problem is ... I can't figure out which factor module, and how to describe it since I rarely work with two indeterminants. I'm still working on part (2).

Anyone can give direction? Thanks for the help.

Note : A module $M$ is indecomposable if its direct summands are only $0$ and $M$.

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  • $\begingroup$ Hint for (2): $\mathbb{Z}$ is indecomposable; is $\mathbb{Z}/6\mathbb{Z}$ indecomposable? $\endgroup$
    – egreg
    Dec 12, 2015 at 18:38
  • $\begingroup$ Thanks for the hint. I'll try it. $\endgroup$
    – user297210
    Dec 13, 2015 at 17:35

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For 1, one good thing to try first is to make the quotient a local ring. That way it is indecomposable as a module, and we could check for decompose or submodules.

So, try doing this with $\frac{\Bbb Q[x,y]}{(x^2,xy,y^2)}$. The ideal generated by $(x,y)$ is the unique maximal ideal...

For 2 you may as well take egreg's advice about choosing $\Bbb Z$ and a quotient.

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