Let $R$ be a ring and consider $R^n$ as an $R$-module. I'm wondering under what circumstances is a submodule of $R^n$ the solution set of some homogeneous system of linear equations in $n$ variables, with variables and coefficients in $R$. I'm particularly interested in the case of finite $R$.
I would appreciate some references on this topic, specifically such that DO NOT assume that the ring is commutative.
First of all, given a system of linear equations in $n$ variables (with coefficients on the left), say
\begin{align*} a_{11}x_1 + \cdots + a_{1n}x_n&=0\\ \vdots&\\ a_{k1}x_1 + \cdots + a_{kn}x_n&=0 \end{align*}
where the $a_{11},...,a_{kn}\in R$, the set of solutions to the equations always form a right $R$-submodule of $R^n$.
If $R$ is a division ring, every submodule of $R^n$ is of this form. Otherwise, this is not true in general. Consider $n=1$, i.e. those right $R$-submodules of $R$, i.e. the right ideals of $R$. Systems of linear equations in one variable take the form $a_ix=0$ for some $a_1,...,a_k\in R$. Therefore, right submodules of $R$ that are solutions to a system of linear equations are those of the form $\operatorname{ann}(I)$ where $I$ is a finitely generated left ideal. Note that in an integral domain, the only ideal arising this way is the zero ideal.
The interesting case, therefore, is when $R$ is a ring for which all right ideals are annihilators of finitely generated left ideals. It would be reasonable to ask, in this case, if right $R$-submodules of $R^n$ are just solution sets of systems of linear equations. I'm not sure either way right now. For examples of rings with this property, it includes all quasi-Frobenius rings (see this) and for some finite examples of quasi-Frobenius rings, all of the rings $\mathbb{Z}/n\mathbb{Z}$ are quasi-Frobenius.