Unit-ideal identity for powers ($(f_1,\dots ,f_r)=R⟹(f_1^N,\dots ,f_r^N)=R$)

If ${\sum }_i{g}_i{f}_i=1$, 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.