# Computing the invariant factors and elementary divisors of finitely generated module over a PID

I have just encountered this question in my abstract algebra class dealing with finitely generated modules over PIDs stating the following:

Let $D = \mathbb{R}[x]$ be the ring of polynomials over the reals in variable x, and let M be a D module, with elements $v_1,v_2,v_3,v_4 \in M$ such that $M = Dv_1 \oplus Dv_2 \oplus Dv_3 \oplus Dv_4$, satisfying: $ann(v_1) = (x-1)^3$ ; $ann(v_2) = (x^2+1)^2$ ; $ann(v_3) = (x-1)(x^2+1)^4$ ; $ann(v_4) = (x+2)(x^2+1)^2$

We are now asked to find the invariant factors and elementary divisors of M

Now all I know is that M is a finitely generated cyclic module over a PID as the direct sum of cyclic modules and I know the fundamental theorems on finitely generated modules over PIDs but I do not know how to extract the invariant factors and elementary divisors from the statement above, so I am hoping for some help on this with a bit of explanation, thanks all helpers

• Do you know how to do this for finitely generated modules over $\mathbb Z$, (abelian groups)? – John Brevik Jan 8 '16 at 1:30
• @JohnBrevik No I do not as I have never extracted this information in the past and do not know how to do this – kroner Jan 8 '16 at 1:32
• I think that would be the right place to start. Whatever text you are using must do this somewhere. – John Brevik Jan 8 '16 at 1:33
• I use Dummit and Foote's Abstract Algebra but did not find this – kroner Jan 8 '16 at 1:35
• Look in Section 5.2 :) – John Brevik Jan 8 '16 at 1:37

you are given a cyclic decomposition (one generator for each summand). the elementary divisors arise from a cyclic, prime power, decomposition. hence you only have to factor all your anihilators into prime powers. That gives $$x+2$$, $$(x-1)$$, $$(x-1)^3$$, $$(x^2+1)^2$$, $$(x^2+1)^2$$, $$(x^2+1)^4$$. Now to get the invariant factors arrange these as relatively prime products in the only possible way so they successively divide each other. That is, $$(x^2+1)^2$$, $$(x-1)(x^2+1)^2$$, $$(x+2)(x-1)^3(x^2+1)^4$$. voila'.