Let $R$ be a finite commutative ring with $1$. Assume that $u$ is a $5$th root of unity in $R$ such that $u - 1$ is a unit in $R$.
From the following system,
$x_1 = x_2 + x_3 + x_4 + x_5, \\ ux_1 = u^3x_2 + u^2x_3 + u^4x_4 + x_5 \\ u^2x_1 = ux_2 + u^4x_3 + u^3x_4 + x_5 \\ u^3x_1 = u^4x_2 + ux_3 + u^2x_4 + x_5 \\ u^4x_1 = u^2x_2 + u^3x_3 + ux_4 + x_5 $
Can anyone show that the solution of this system over the ring $R$ is unique which is $(0,0,0,0,0)$?
I have already checked that this system has only one solution which is $(0,0,0,0,0)$ if we work over $\mathbb{R}$ or $\mathbb{C}$.