Let $\mathfrak{a}$ and $\mathfrak{b}$ be ideals of a ring $R$ Denote by $\mathfrak{ab}$ , the subset of $R$ formed by all finite sums of products $ab$ with $a \in \mathfrak{a}$ , and $b \in \mathfrak{b}$ It is straightforward to verify the following facts:

• If $\mathfrak{a}$ is a left and $\mathfrak{b}$ a right ideal, $\mathfrak{ab}$ , is a two-sided ideal of $R$
• If both $\mathfrak{a}$ and $\mathfrak{b}$ are two-sided ideals, then $\mathfrak{ab} \subseteq \mathfrak{a}\cap\mathfrak{b}$