I am learning about the linear system of Diophantine equations. I have noticed that the solution structure of the Diophantine equation is quite similar to the general system of linear equations. It is $S=x_0+F$, where $x_0$ is a partial solution and $F$ is the solution set of the corresponding homogeneous Diophantine linear system, $F$ is the submodule of $\mathbb{Z}$-module $\mathbb{Z^n}$.
But in the general system of equations, $F$ is a subspace of the vector space $K^n$.
So, is the proof for the root structure theorem of the linear system of the Diophantine equations similar to the general system of equations? If it is different, can someone give me a tutorial to proof, please?
My proof for the structure of the solution set of a general linear equation system:
Theorem: If the general system of linear equations $\left\{\begin{array}\ a_{11}x_1+\cdot+a_{1n}x_n=b_1\\ \vdots\\ a_{m1}x_1+\cdots+a_{mn}x_n=b_m \end{array}\right. (*) $ has a solution, then the solution set $S$ has the form $$S=x_0+F,$$ where $x_0$ is a partial solution of $(*)$ and $F$ is the solution set of the corresponding homogeneous system of linear equations, $F$ is a subspace of $K^n$.
Proof: The system $(*)$ can be written as $f(x)=b$. We have $f(x_0)=b$ and $F=Ker(f)$. Suppose that $x\in K^n$ is any solution of $(*)$, then $f(x)=b$. This implies $f(x-x_0)=0$. Hence $x-x_0\in Ker(f)=F$. Thus $x\in x_0+F$. Conversely, if $x=x_0+w$, for $w\in F$, then $$f(x)=f(x_0+w)=f(x_0)+f(w)=b+0=b.$$ Therefore $x$ is a solution of $(*)$. Hence $S=x_0+F$.
