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Homework answers / question archive / Let phi:R->S be a homomorphism of commutative rings a) Prove that if P is a prime ideal of S then either phi^-1(P)=R or phi^-1(P) is a prime ideal of R

Let phi:R->S be a homomorphism of commutative rings

a) Prove that if P is a prime ideal of S then either phi^-1(P)=R or phi^-1(P) is a prime ideal of R. Apply this to the special case when R is a subring of S and phi is the inclusion homomorphism to deduce that if P is a prime ideal of S then PR is either R or prime ideal in R

b) Prove that if M is a maximal ideal of S and phi is surjective then phi^-1(M) is maximal ideal of R. Give an example to show that this need not be the case if phi is not surjective.

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