Soft question about integral domain

Question

Here is a soft question that I am dealing with. Please tell me if it's correct or not.

Suppose $A$ is a commutative ring with unity. Is $A$ a prime ideal of $A$?

I think the answer is true, because we know $I$ is an integral domain iff $R/I$ is an integral domain. But $A/A = 0.$ So it's an integral domain. So$A$ is a prime ideal in $A$ Tell me if I argument is right or wrong. Any help will be appreciated. Thanks

Answer 1

It's usually taken as part of the definition that an integral domain must not be the zero ring.

For example, on Wikipedia (first sentence of the article, note the word “nonzero”), and in the standard references (Atiyah & Macdonald, Matsumura, Lang, etc.).