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