Let ∫ : R → S be an epimorphism of rings with kernel K.

(a) If P is a prime ideal in R that contains K, then f(P) is a prime ideal in S [see Exercise 13].

(b) If Q is a prime ideal in S, then ∫^-1( Q) is a prime ideal in R that contains K.

(c) There is a one-to-one correspondence between the set of all prime ideals in R that contain K and the set of all prime ideals in S, given by P|→ ∫(P).

(d) If I is an ideal in a ring R, then every prime ideal in R/ I is of the form PI, where P is a prime ideal in R that contains I. 

Data from exercise 13

Let R be a ring without identity and with no zero divisors. Let S be the ring whose additive group is R X Z as in the proof of Theorem 1.10. Let A= {(r,n) ϵ S|rx+nx = O for every x ϵ R}.

(a) A is an ideal in S.

(b) S/ A has an identity and contains a subring isomorphic to R.

(c) S/ A has no zero divisors.

I, where P is a prime ideal in R that contains I.