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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 $R_1\times \{0\}$.

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If you do not impose other conditions, there are in general an immense number of ring structures with that underlying additive group.

For example, let $R_1=R_2=\mathbb R$. Then $R_1\times R_2$ is in fact isomorphic to $\mathbb R$ as an abelian group, and there are uncountably many isomorphisms $R_1\times R_2\to\mathbb R$. From each of them you get a ring structure.

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