# Example of a ring with prime characteristic, which is not an integral domain

We know that every intgral domain has prime (or 0) characteristic. Is there en example that the converse isn't true? Does there exist a ring, which is not an integral domain, but has a prime characteristic?

$\frac{\mathbb Z}{\mathbb Zp}\times\frac{\mathbb Z}{\mathbb Zp}\,.$ For the characteristic $0$ case, consider $\mathbb R\times\mathbb R\,.$
Yes, of course. Take $\mathbb{F}_p[t]/(t^2)$.
• $\mathbb{F}_p[t]$ are polynomials over given field $\mathbb{F}$ that has p elements, like $\mathbb{Z}_p$? – user19502 Oct 22 '13 at 1:03
• @user19502: Yes, exactly. $\mathbb{F}_p$ is another notation for the field $\mathbb{Z}/p\mathbb{Z}$. – Najib Idrissi Oct 22 '13 at 13:13