Unit-ideal identity for powers
= Unit-ideal identity for powers
{title2=$(f_1,\dots,f_r)=R\Longrightarrow(f_1^N,\dots,f_r^N)=R$}
If $\sum_i g_if_i=1$, expand its power $r(N-1)+1$. By the <pigeonhole principle>, each monomial contains some $f_i^N$. Collecting terms expresses $1$ as a <linear combination> of these powers. This lets one use a common denominator exponent in <localization> arguments.