A question about invariant factors

[Artusartos]

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

 

[mathwonk]

well you have to be given something. i have some examples in my book on my webpage.

to be "given" a f.g. module usually means to be given a "presentation" as a quotient of two free modules.

such a quotient is specified by a matrix. then you diagonalize that presentation matrix.

see the discussion here:

http://www.math.uga.edu/%7Eroy/845-1.pdf