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: $\mathrm{ann}(v_1) = (x-1)^3 $; $\mathrm{ann}(v_2) = (x^2+1)^2$; $\mathrm{ann}(v_3) = (x-1)(x^2+1)^4$; $\mathrm{ann}(v_4) = (x+2)(x^2+1)^2 $. We are now asked to find the invariant factors and elementary divisors of $M$. (Jacobson, Basic Algebra I, Exercise 3.9.1)
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