Can 0 be a maximal ideal or prime ideal of R? If this so, what does it say about R in either case ?

This is my work so far

Claim: 0 is a prime ideal if R is an integral domain

Because 0 is a prime if and only if 0|ab implies 0|a or 0|b

That is : ab = 0 ----> a = 0 or b = 0

Hence R is an integral domain

Is that correct?

And 0 is a maximal ideal. I am not really sure about this. I think the answer is yes if we say R = Q (rational numbers) but I don't know how to explain.

Thank you very much