It seems a typical exercise in module theory is to write a finitely generated module over a PID as a direct sum of cyclic modules. After doing several problems, I have come to the conclusion that there are only finitely many variations of this type of problem.
Typically there are three standard ways this question can be asked:
(1) A matrix associated to the homomorphism $\phi$ in a free resolution for $M$, $0 \rightarrow R^n \xrightarrow{\phi} R^m \rightarrow M \rightarrow 0$, is given and we can use the smith normal form of the matrix to write $M$ as a direct sum of cyclic modules.Example:
Describe, as a direct sum of cyclic groups, the cokernel of the map $\phi: \mathbb{Z}^{3} \longrightarrow \mathbb{Z}^{3}$ given by left multiplication by the matrix
$$ \left(\begin{matrix} 15 & 6 & 9 \\ 6 & 6 & 6 \\ -3 & -12 & -12 \end{matrix}\right) $$
(2) A homomorphism $\phi$ between free modules is given. A list of generators for the image of $\phi$ is given. One can insert the vector corresponding to each generator into a matrix and find the smith normal form, which allows one to write $M=\text{coker} \phi$ as a direct sum of cyclic modules.Example:
Let $G$ be the quotient group $G=\mathbb{Z}^5/N$, where $N$ is generated by $(6,0,-3,0,3)$ and $(0,0,8,4,2)$. Recognize $G$ as a product of cyclic groups.
(3) (This problem type I understand the least) Relations for generators of $M$ are given. For example if $M$ is generated by $x_1,...,x_n$ subject to the relations $a_{11}x_1+ \cdots +a_{1n}x_n=0,...,a_{n1}x_1+\cdots +a_{nn}x_n=0$, one can row reduce the matrix corresponding to this system of equations, to a smith normal form, and write down the cokernel of the homomorphism represented by the matrix in a similar manner to the above two.( If someone has a good source on relations for generators of finitely generated modules I would greatly appreciate it)Example
Let M be the module over Z [i] generated by elements x, y whose relations are determined by $(1+i)x+(2-i)y=0$ and $3x+5y=0$. Write M as a direct sum of cyclic modules.
I was given the following problem:Let $R=\mathbb{Q}[x]$ and consider the submodule $M$ of $R^2$ generated by the elements $(x^2-1,x-1)$ and $(x^2+x,x)$. Write $M$ as a direct sum of cyclic submodules. At first I blindly attempted to insert these vectors into a matrix and compute the smith normal form to do this. Then I was told, these are generators of $M$, not the image of the corresponding homomorphism, with $M$ the cokernel. So this strategy is not approperiate.
I get that this problem can be solved in a simple manner as follows
Since $$(x+1,1)=x(x+1,1)-(x-1)(x+1,1)\in M$$ and the two generating elements are scalar multiples of this element, this implies every element of $M$ is a scalar multiple of $(x+1,1)$. Thus, $M=\langle (x+1,1) \rangle$
However, what if we are given a more complicated collection of generators for $M$? Is there a way to find the the relations for these generators? Then can one insert these into a matrix, find the smith normal form, and write $M$ as the direct sum that way? How would one go about doing that for this problem?
Also what other types of iterations of writing a finitely generated module over a PID as a direct sum of cyclics, are out there, other than the ones I listed?