The direct product of two rings $R$ and $S$ is the ring $R \times S$ with addition and multiplication defined by
\[\begin{align*} (r_1, s_1) +(r_2, s_2) &= (r_1 + r_2, s_1+s_2), \\ (r_1, s_1)(r_2, s_2) &= (r_1 r_2, s_1 s_2) \end{align*}\]for all $r_1$, $r_2 \in R$ and $s_1$, $s_2 \in S$.