# 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 at 17:31
• Near-duplicate: math.stackexchange.com/questions/1326505/… – Eric Wofsey Aug 19 at 17:58
• Also relevant: math.stackexchange.com/questions/427078/is-0-a-field – Eric Wofsey Aug 19 at 18:01
• So mostly it deoends on the convention. But I don't understand the argument behind whole ring not being a prime ideal. In the questions they discussed about 0 being any domain, while not about 0 being an integral domain. So I should consider 0 in general not an integral domain and the proceed further? – user631697 Aug 19 at 18:13
• That's not a soft question. – Paul Aug 19 at 20:01