Finding generators for products of ideals If you want to find the generators for the product of ideals, do you simply take all possible products of the generators in the ideals. For example, let $R$ be a ring and let $I = (a,b)$ and $J = (c,d)$. Then we have
\begin{equation*}
\begin{aligned}
IJ &= \{(ax + by)(cw + dz) : w,x,y,z \in R\} \\
&= \{ac(xw) + ad(xz) + bc(yw) + bd(yz) : w,x,y,z \in R\}.
\end{aligned}
\end{equation*}
We see that $IJ$ contains all multiples of $ac,ad,bc,bd$ and their sums. So then $IJ = (ac,ad,bc,bd)$. Is this true in general?
Thanks!
 A: 
Does this method work regardless of the number of ideals and
  generators is what I mean.

Yes, it does. Let $I = (a_1, \ldots, a_m)$ and $J = (b_1, \ldots, b_n)$ be left ideals in a ring commutative ring $R$. Then
$$IJ = \{(x_1a_1 + x_1a_2 + \ldots + x_ma_m)(y_1b_1 + y_2b_2 + \ldots + y_nb_n) \}$$
and clearly $a_ib_j \in IJ$ for all $1 \le i \le m$ and $1 \le j \le n$. Conversely, every element in $IJ \setminus \{0\}$ is of the form $\sum_{i,j} z_{i,j}a_i b_j$.
An easy induction on the number of factors yields the result for $I_1I_2 \ldots I_n$ as well.
However, in a non-commutative ring we only get $(a_1, \ldots, a_m)(b_1, \ldots, b_n) \supseteq (a_ib_j)_{i,j}$.
As an example consider the ring of $2 \times 2$ matrices with integer coefficients. Then 
$$
\left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right) \left( \begin{matrix} 0 & 0 \\ 1 & 1 \end{matrix} \right) = \left( \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} \right)
$$
and thus $ \left( \left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right)  \left( \begin{matrix} 0 & 0 \\ 1 & 1 \end{matrix} \right) \right) = (0) $, but
$$
\left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right)\left( \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix} \right) \left( \begin{matrix} 0 & 0 \\ 1 & 1 \end{matrix} \right) =  \left( \begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix} \right)
$$
demonstrates $ \left( \left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right),  \left( \begin{matrix} 0 & 0 \\ 1 & 1 \end{matrix} \right) \right) \neq (0)$.
