A question on generators

Question

Suppose $I$ is an ideal in a ring $R$ which is finitely generated. Suppose on the other hand that there is some (possibly other) set of generators $\{g_t\colon t\in T\}\subset I$ which also generates $I$ as an ideal. Can we find a finite subset $T_0\subset T$ such that $\{g_t\colon t\in T_0\}$ generates $I$?

Answer 1

$I$ is generated by $(f_1, \ldots, f_n)$ as per your first assumption. Since the family of $(g_t)_{t\in T}$ generates $I$, for all $i$ we have $f_i = \sum a_{i,t} g_t$, the sum being finite. Take $T_0$ to be the set of all $g_t$'s that appear in those $n$ summations, $T_0$ is finite and still generates $(f_1,\ldots, f_n)$ so it generates $I$.