The notation I am using is from N. Jacobson Algebra I. I am trying use his method in 3.6 of determining a basis for a sub-module in order to determine the cyclic decomposition of a finitely generated module.
I will go through my reasoning and solution for the problem I am looking for input on if I made any mistakes.
Problem statement: First find a basis for $\mathbb{Z}^{(3)}$ consisting of the set of all $(x,y,z)$ satisfying $2x+6z=0$ and $4x+8y=0$ and determine the structure of the sub-module as a sum of cyclic groups.
First we determine a basis for the submodule by forming the matrix $ \left( \begin{array}{ccc} 2 & 0 & 6\\ 4 & 8 & 0 \end{array} \right) $ After some elementary row operations we get $ \left( \begin{array}{ccc} 1 & 2 & 0\\ 0 & 2 & -3 \end{array} \right) $ which tells us our basis must satisfy $x = 2y = -3z$ and $2y + 3z =0$.
Question 1. Does this tell us $<1,2,0>$ and $<0,2,-3>$ form a basis for our sub-module?
Question 2. Since we can generate $x$ from $y$ and $z$ and since $2y + 3z =0$ does it follow that the cyclic subgroup decomposition of the sub-module generated by $x,y,z$ is $\mathbb{Z} \oplus \mathbb{Z_2} \oplus \mathbb{Z_3}$
Thanks for all your help.