(a) Let R be an integral domain with quotient field F. If O a R,...

(a) Let R be an integral domain with quotient field F. If O ≠ a ϵ R, then the following are equivalent:

(i) Every nonzero prime ideal of R contains a;

(ii) Every nonzero ideal of R contains some power of a;

(iii) F = R[1_R/ a] (ring extension). An integral domain R that contains an element a ≠ 0 satisfying (i)-(iii) is called a Goldmann ring.

(b) A principal ideal domain is a Goldmann ring if and only if it has only finitely many distinct primes.

(c) Is the homomorphic image of a Goldmann ring also a Goldmann ring?