TITLE: About generators of a finitely generated ideal
QUESTION [5 upvotes]: Let $R$ be a ring with $1$. Let $S$ be a subset of $R$, with infinitely many elements. Let $\mathfrak{i}$ be the ideal of $R$ generated by $S$. Suppose $\mathfrak{i}$ finitely generated:
$$\mathfrak{i}=Rr_1+\ldots+Rr_n$$
with $r_i\in R$. Can i always find generators $r_i$'s in $S$?

REPLY [5 votes]: Yes you can. Suppose $I=\lbrace a_1 r_1+\cdots +a_nr_n|a_i\in R\rbrace$, now each $r_i\in I$ implies one can write $r_i=\sum_{j=1}^{n_i} a_{i_j}s_{i_j}$, where $s_{i_j}\in S$. Then we have $I\subseteq\lbrace \sum_{i=1}^ma_i s_i|a_i\in R\rbrace :=J$ for some finite number of $s_j$'s. Also note that $J\subseteq I$, because $I$ is generated by $S$ and so $s\in I$ for each $s\in S$.