If $R$ and $S$ are fields, either prove or disprove that $R\times S$ is a field
That's the question from my homework. I am thinking $R\times S$ is not a field, but I'm not sure. I understand the definition of a field, but I am not sure how to proceed.
Hint: Every non-zero element in a field has a multiplicative inverse. Can you find a non-zero element in the product of two fields that has no inverse?
$R \times S$ is never a domain, even if $R$ and $S$ are domains because $(1,0)\cdot(0,1)=(0,0)$ shows that $R \times S$ has zero divisors.
Hint: Use the characterization of a field in terms of its ideals. What are the ideals of $R \times S$?