# Soft question about integral domain

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

• Usually when we consider prime ideals, we define them to be proper ideals, i.e. ideals that are not the whole of $A$. – rubikscube09 Aug 19 '19 at 17:31
• Near-duplicate: math.stackexchange.com/questions/1326505/… – Eric Wofsey Aug 19 '19 at 17:58
• Also relevant: math.stackexchange.com/questions/427078/is-0-a-field – Eric Wofsey Aug 19 '19 at 18:01
• That's not a soft question. – Paul Aug 19 '19 at 20:01
• @user631697 you could call it a convention, but if so, this is really one of those conventions that has nearly 100% of the usage. I have never seen the alternative used even once in a text or paper. – rschwieb Aug 20 '19 at 2:35