The product of two finitely generated ideals is finitely generated

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.12
Solution:
($\subseteq$) Let $x = \sum_i r_is_i \in IJ$, where $$r_i = \sum_j t_{i,j}a_j$$ and $$s_i = \sum_k u_{i,k} b_k.$$ Then we have $$x = \sum_i (\sum_j t_{i,j}a_j)(\sum_k u_{i,k}b_k) = \sum_i \sum_j \sum_k t_{i,j}u_{i,k}a_jb_k \in (C)$$ since $R$ is commutative.
($\supseteq$) Let $x = \sum r_{i,j}a_ib_j \in (C)$. Since $I$ is an ideal, $x \in IJ$.