Let $\displaystyle B=A_{1}\times\cdots\times A_{n}$ be a product of rings. Any prime ideal of B is of the form $\displaystyle A_{1}\times\cdots\times I_{i}\times\cdots \times A_{n}$, where $\displaystyle i\in\{1,\cdots,n\}$ and $\displaystyle I_{i}$ is a prime ideal of $\displaystyle A_{i}$.

Thanks in advance.