Math Help - Checking answer: is 0 a maximal ideal?

1. Checking answer: is 0 a maximal ideal?

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

2. Originally Posted by knguyen2005
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

Your claim is correct, and about maximal: {0} is a maximal ideal of R iff $R/\{0\}$ is a field, but...

Tonio

3. Originally Posted by tonio
Your claim is correct, and about maximal: {0} is a maximal ideal of R iff $R/\{0\}$ is a field, but...

Tonio
About the maximal ideal,
So, {0} is a maximal ideal of R iff $R/\{0\}$ is a field. Then we can take R = IR, Q(rational number), C (Complex number). Is that what you are saying.

Thanks Tonio

4. Originally Posted by knguyen2005
About the maximal ideal,
So, {0} is a maximal ideal of R iff $R/\{0\}$ is a field. Then we can take R = IR, Q(rational number), C (Complex number). Is that what you are saying.

Thanks Tonio

What I was saying is that $R/\{0\} \sim R$, and thus R is a field.
It can be of characteristic zero, as the ones you mentioned, of a finite or infinite field of finite positive characteristic.

Tonio