Answer to Question #201794 in Abstract Algebra for Komal

Question

Show that pZ is a prime ideal of Z for all primes p.

Answer 1

We can use the fact that if an ideal I of R is prime, then R/I is an integral domain.

So we let Z be our ring, and I be any ideal of Z. The claim is that I=(p) where p is prime.

Proof:

If I is a prime ideal, then Z/I is an integral domain. Since all ideals of Z are principle, Z/(n)=Z_n. Since I is prime, Z_n must be an integral domain. If it is an integral domain, it must also be a field, since it is finite integral domain. But the only finite fields of the form Z_n are those with n being a prime number. Therefore, all prime ideals are of the form (p).