Ideal generated in a noetherian ring

Question

I read a proof of a theorem but there is a part that I don't quite understand: If $A$ is a Noetherian Ring and if $J$ is an ideal of $A$ of finite type generated by $(a_i)_{i\in I}\in A^I$, then there exist $i_1, i_2,\ldots, i_n\in I$ such that J is generated by $a_{i_1}, a_{i_2},\ldots,a_{i_n}$

Thanks in advance.

P.S. I know that any Ideal of a Noetherian Ring is of finite type.

Answer 1

Let $(b_1,\dots, b_r)$ be a finite set of generators for $J$. Each of these generators is a finite linear combination of elements in $(a_i)_{i\in I}$. For each $j\enspace(1\le j\le r)$, denote $S_j$ the set of the $i$s in $I$ involved in an expression of $b_j$ as a linear combination of elements in $(a_i)_{i\in I}$. Then the set $$G=\bigl\{a_i\mid i\in S_1\cup\dots\cup S_r\bigr\}$$ is a finite set of generators for $J$, contained in the initial set $(a_i)_{i\in I}$.