I got the question below studying this problem: $p$-Sylow subring.
Let $(R_1,+_1,\cdot_1)$ and $(R_2,+_2,\cdot_2)$ be two rings with identity elements $e_1,e_2$.
Let $(R,+)$ be the group defined by $R=R_1\times R_2, \quad (a,b)+(c,d)=(a+_1c,b+_2d).$
Question: how may products $(\cdot)$ can we define on $R$ in a way that $(R,+,\cdot)$ could be a ring?
We know that we can define $(a,b)\cdot (c,d)=(a\cdot_1c,b\cdot_2d)$ and we obtain an identity element $e=(e_1,e_2)\in R$. But note that this element is not the same as the element $(e_1,0)$, which is an identity for $R_1\times \{0\}$.