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