referring to the integers... sorry, I am not sure how to type-face it as bold.
Is every prime ideal of also a maximal ideal?
If so, how can this be justified?
If not, can anyone think of example?
Obviously, every maximal ideal is also prime. Also, if we were in an finite integral domain, then every prime ideal would be maximal... but has zero divisors, so we are not in an integral domain, and it is also infinite.
Any help much appreciated!
Thanks.
Every ideal of a finite product of rings is a product of ideals in each ring. Such an ideal is prime if and only if it is a product of a prime ideal in one factor and the entire ring in each other factor. Since prime ideals of Z are maximal, the answer is yes.
No, you are correct.
It is true that is an ideal if and only if , each an ideal of . Also, is prime implies each is prime in or equals ; however, the converse is not necessarily true (e.g., ).
I don't believe anything can necessarily be said about maximal ideals.
In this case we have (where the first two run over all primes). The latter two are not gonna be maximal, as you have noticed. The others are all maximal, though.