Let R be a PID and P a fixed prime ideal of R satisfying 0r

Let R be a PID and P a (fixed) prime ideal of R satisfying (0_r) P (1_r). Say P = (p) with p R. Prove that p is a prime element of R. True or false: p is an irreducible element of R. True or false: P is a maximal ideal of R. Could there ever exist a prime ideal Q such that (0_r) Q P? Explain why.

Solution

1.

An element p of a commutative ring R is said to be prime if it is not zero or a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b. Equivalently, an element p is prime if, and only if, the principal ideal (p) generated by p is a nonzero prime ideal.^[1] (Note that in an integral domain, the ideal (0) is a prime ideal, but 0 is an exception in the definition of \'prime element\'.)

2. TRUE

3. TRUE

4. NO. Every prime ideal of a principal ideal domain is maximal ideal.