What does it mean here to describe the structure of this quotient module? I'm working on a problem asking me to determine the structure of $\mathbb{Z}^3/K$ where $K$ is generated by $(2,1,-3)$ and $(1,-1,2)$. I suspect as a $\mathbb{Z}$-module.
My first guess was that $\mathbb{Z}^3/K$ is isomorphic to some finitely generated $\mathbb{Z}$-module $M$ with generators $x_1,x_2,x_3$ maping to the standard basis elements $e_1,e_2,e_3$, but that was hard to work with.
I then set up a matrix
$$
\begin{pmatrix} 2 & 1 & -3 \\ 1 & -1 & 2\end{pmatrix}
\sim
\begin{pmatrix} 1 & 0 & 0 \\ 0 & 3 & 0\end{pmatrix}.
$$
Does this just mean $\mathbb{Z}^3/K\cong \mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z}$? Is this the correct thing to do, if so, why does it work?

The matrix operations are equivalent to quotient module isomorphisms. 
[Actually row operations just replace relations with equivalent relations. Column operations require isomorphisms.]
Thus you have $\mathbb{Z}^3/K \cong \mathbb{Z}^3/L$ where $L$ is generated by $(1,0,0)$ and $(0,3,0)$.
So your new relations say 
$(1,0,0)=(0,0,0)$  
and 
$(0,3,0)=(0,0,0)$  
(mod 1 in the first coord and mod 3 in the second).
Therefore, your module is isomorphic to $\mathbb{Z}/3\mathbb{Z} \oplus \mathbb{Z}$.
