Why does $\langle a \rangle \langle b \rangle \subset \langle ab \rangle$ in a commutative ring?

Question

My friend and I saw this affirmation in a proof and we're having trouble figuring out why is this valid.

Thanks in advance!

Sorry: forgot to add some details. Consider a commutative ring A and an ideal prime I of A. We have $a,b \in I$.

Answer 1

By definition, we have elements of $(a)(b)$ are finite sums of the form $$\sum r_ias_ib$$ for some $r_i,s_i\in A$. By commutativity, we have $$\sum r_ias_ib=\sum r_is_iab=\sum l_iab$$ for some $l_i\in A$.

Then $\sum l_iab\in (ab)$ clearly.

We don't need primeness of $I$.