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I'm currently taking a course in group- and ring theory, and we are now dealing with a chapter on finitely generated modules over PIDs. I have stumbled across some problems that I can't really get my head around. It is one in particular that I would very much like to understand, and I would greatly appreciate some help.

The situation is the following. Assume we are given a free module, for example the [tex]\mathbb{Z}[/tex]-module [tex]\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}[/tex], and want to consider the submodule generated by all elements [tex]\left(x_{1},x_{2},x_{3}\right)[/tex] satisfying certain relations [tex]\sum_{1}^{3}{a_{i,j}x_{j}} = 0[/tex] for [tex]i=1,2[/tex]. I have heard that a good way of understanding this submodule would be to consider the matrix [tex](a_{i,j})_{2,3}[/tex], put it in smith normal form and then conclude that the entries on the generalized diagonal will be the torsion coefficients in the decomposition of the submodule as a direct sum of cyclic ones (as in the structure theorem). However, I have not managed to draw that conclusion. If someone could explain why this holds true, or if it is false tell me that (and perhaps give me a hint on how to understand said submodule), I would be extremely thankful.

Thanks in advance!

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# Finitely generated modules over a PID, and applications on abelian groups

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