I confronted with a statement:
$A$ is a commutative ring, and $f_1,f_2,...,f_r$ generate the unit ideal, then $f_1^N,f_2^N,...,f_r^N$ also generate unit ideal for any positve integer $N$.
I have no idea how to proof this statement. On the contrary, it seems that I can give a counterexample. Consider ring $Z[x]$, then $f_1=x,f_2=x+1$ generate unit ideal $(1)$, but $f_1^2=x^2,f_2^2=(x+1)^2=x^2+2x+1$ can't generate unit idea! I hope someone can help me out. Thanks!
Your example is incorrect, since $$ (x^2+2x+1)(x^2-2x+1) - x^2 (x^2 - 2) = 1. $$
To show that $(f_1^N,\dots,f_n^N)$ generates the unit ideal, raise the equation $$ f_1 g_1 + \dots + f_n g_n = 1 $$ to the $(N-1)n+1$-th power, and note that every term must have at least an $f_i$ that is raised to power $\geq N$ by pigeonhole.