Let $\{f_i\}$ be a system of linear equations in $X_1,...,X_n$ with coefficients in $R = \mathbb Z/p^e \mathbb Z$ (i.e. modulo a primepower).
Assume
- there is a unique solution $a_i$.
- $f_1 = u g_1$ where $u$ is a non-unit, i.e. $p\mid u$.
Is it true that $\{f_i\mid i>1\}$ permits only the same unique solution? Or equivalently: is $f_1$ linearly dependent on the other equations?
Intuitively, it seems obvious. For if we substitute variables such that $g_1 = (Y_1 - a)$ then $Y_1$ is determined by $f_1$ up to some multiple of $p$. But there's no way the other equations could determine what multiple without determining what $Y_1$ is itself, since the only non-units are all multiples of $p$. But that is, of course, not very solid mathematics.