Domain prime localizations do not imply a domain
= Domain prime localizations do not imply a domain
{title2=$A=k\times k$}
The <ring> $k\times k$, for a <field> $k$, has two <prime ideals>, $0\times k$ and $k\times0$. Their <localizations at a prime ideal> are both isomorphic to $k$. Nevertheless $(1,0)(0,1)=0$ is a product of nonzero elements, so $A$ is not an <integral domain>. Nonzero orthogonal <idempotents> separate the two components of its <spectrum of a commutative ring>.