# A question about invariant factors

1. Jan 16, 2013

### 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?

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Last edited: Jan 16, 2013
2. Jan 18, 2013

### 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 [Broken]

Last edited by a moderator: May 6, 2017