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