Noetherian rings/Hilbert's Basis Theorem

Question

So I'm studying the proof of Hilbert's Basis Theorem - we've shown that $λ(I)$ is an ideal of $R$ and and then it says "Since $R$ is Noetherian, we have $λ(I) = \sum\limits_{i=1}^k s_iR$ for some $s_1, ..., s_k \in R$". I don't see where this comes from - is it a standard fact about ideals and Noetherian rings? Thanks!

Answer 1

$R$ is a Noetherian ring if and only if every ideal in $R$ is finitely generated.

Since you've shown that $\lambda(I)$ is an ideal of $R$, it is finitely generated, that is, $$\lambda(I) = \langle s_1, \ldots, s_k \rangle = \sum_{i = 1}^k s_iR$$ where $s_i \in R$.