Prove that every maximal ideal in an integral domain is also

Prove that every maximal ideal in an integral domain is also a prime ideal.

Solution

Notice that for principal ideals: contains=divides , i.e. (a)(b)ab. Therefore,

(p) is maximal (p) has no proper container (a)

                     p   has no proper divisor a

                     p   is irreducible

                      (p)  is prime,  by PIDUFD, so ireducible = prime

PIDs are the UFDs of dimension 1, i.e. where all prime ideals 0 are maximal.