## Representativewise multiplication of cosets of an ideal is well-defined

Let $R$ be a ring and let $I \subseteq R$ be an ideal. Show that if $a_1 - b_1 \in I$ and $a_2 - b_2 \in I$, then $a_1a_2 - b_1b_2 \in I$.

Since $I$ is an ideal, $a_1a_2 - b_1a_2 \in I$ and $b_1a_2 - b_1b_2 \in I$. So $a_1a_2 - b_1a_2 + b_1a_2 - b_1b_2$ $= a_1a_2 - b_1b_2 \in I$ as desired.