Problem 27. Let R, S be rings and let y: R→ S be a surjective ring homomorphism. Let p be a prime ideal in R with ker(y) CP. (a) Prove that y(p) is a prime ideal in S. (b) Prove that if q is a prime ideal in S, then y¹(q) is a prime ideal in R with ker(y) < y¹(q). (c) Let Spec(S) be the set of prime ideals in S and let Spec(R; y) be the set of prime ideals p in R with ker(y) C p. Define Y: Spec(S) →→Spec(R; v) def by Y(q)-¹(q). Prove that is bijective.
Problem 27. Let R, S be rings and let y: R→ S be a surjective ring homomorphism. Let p be a prime ideal in R with ker(y) CP. (a) Prove that y(p) is a prime ideal in S. (b) Prove that if q is a prime ideal in S, then y¹(q) is a prime ideal in R with ker(y) < y¹(q). (c) Let Spec(S) be the set of prime ideals in S and let Spec(R; y) be the set of prime ideals p in R with ker(y) C p. Define Y: Spec(S) →→Spec(R; v) def by Y(q)-¹(q). Prove that is bijective.
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Transcribed Image Text:Problem 27. Let R, S be rings and let y: R→ S be a surjective ring homomorphism. Let p be a prime ideal in
R with ker(y) CP.
(a) Prove that y(p) is a prime ideal in S.
(b) Prove that if q is a prime ideal in S, then y¹(q) is a prime ideal in R with ker(y) < y¹(q).
(c) Let Spec(S) be the set of prime ideals in S and let Spec(R; y) be the set of prime ideals p in R with
ker(y) C p. Define
Y: Spec(S) → Spec(R; y)
def
by Y(q) ¹(q). Prove that is bijective.
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