## No field is a nontrivial direct product

Prove that if $R$ and $S$ are nonzero rings then $R \times S$ is not a field.

Let $a \in R$ and $b \in S$ be nonzero. Then $(a,0),(0,b) \in R \times S$ are nonzero, but $(a,0)(0,b) = 0$. Thus $R \times S$ contains zero divisors and cannot be a field.