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A Theorem in our textbook says...

If R is a PID, then every finitely generated torision R-module M is a direct sum of cyclic modules

M= R/(c_1) \bigoplus R/(c_2) \bigoplus ... \bigoplus R/(c_t)

where t \geq 1 and c_1 | c_2 | ... | c_t .


There is an example from our textbook that I attached...they find the invariant factors from the elementary divisors. But what if we had to find the invariant factors without being given the elementary divisors. How would we do that?

Thanks in advance
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