Chapter 6.1, Problem 10E

Let $I_1$ and $I_2$ be ideals of the ring $R$. Prove that the set

$I_1{I}_2=\{a_1{b}_1+a_1{b}_2+\mathrm{...}+a_n{b}_n|a_i\in I_1,b_i\in I_2,n\in Z^{+}\}$

is an ideal of $R$. The ideal $I_1{I}_2$ is called the product of ideals $I_1$ and $I_2$.