Let $A,B,C$, and $D$ be integral domains. If $A\times B$ is isomorphic to $C\times D$, prove that $A$ is isomorphic to $C$ or $D$.

Question

Let $A,B,C$, and $D$ be integral domains. If $A\times B$ is isomorphic to $C\times D$, prove that $A$ is isomorphic to $C$ or $D$.

I am doing some practice on my qualifying exam next month. However, not like groups, I haven't seen many problems on direct product of rings, except for the Chinese Remainder theorem. Can anyone help me solve this problem? I actually do not how to start. Thanks.

Answer 1

Since $A$ and $B$ are domains, the ring $A \times B$ contains exactly four idempotent elements, so this must also be true of $C \times D$. Let $e = \phi(1, 0)$, then $e$ is idempotent and can't be zero or the identity, so it must be $(0, 1)$ or $(1, 0)$, so $A$ is isomorphic to either $C$ or $D$ via $\phi$.