Let phi: R --> R' be a homomorphism where both R and R' are commutative rings and I is a subring of R. Show that if J is a prime ideal of R' and I = phi^-1(J), then I is a prime ideal of R.
Let : R --> R' be a homomorphism where both R and R' are commutative rings and I is a subring of R. Show that if J is a prime ideal of R' and I = (J), then I is a prime ideal of R.
is an ideal of because is an ideal of and we don't need any assumptions on (including assuming that is a subring!!).
to show that is prime, just follow the definition again: if then but is a prime ideal of so either or hence either or