Suppose that R and S are commutative rings with unites, Let ϕ be a ring homomorphism from R onto S and let A be an ideal of S. a. If A is prime in S, show that ϕ−1(A)={x∈R∣ϕ(x)∈A} is prime ∈R. b. If A is maximal in S, show that ϕ−1(A) is maximal ∈R.

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Suppose that R and S are commutative rings with unites, Let ϕ be a ring homomorphism from R onto S and let A be an ideal of S. a. If A is prime in S, show that ϕ−1(A)={x∈R∣ϕ(x)∈A} is prime ∈R. b. If A is maximal in S, show that ϕ−1(A) is maximal ∈R.

stuth1 · Accepted Answer

a. Suppose A i sprime ideal in S. Consider ab∈ϕ−1(A). Then ϕ(ab)=ϕ(a)ϕ(b)∈A. Since A is prime, either ϕ(a)orϕ(b) belongs to A. Hence either a or b belongs to ϕ−1(A) Thus, ϕ−1(A) is prime ∈R b. Consider the homomorphism phi: R→SA defined by ϕ(r)=ϕ(r)+A Then the kernel of ϕ is ϕ−1(A). Then,Rkerϕ≈SA Therefore, ϕ−1(A) is maximal ideal.

Suppose that R and S are commutative rings with unites, Let PSJphi be a ring homomorphism from R onto S and let A be an ideal of S.a. If A is prime in S, show that phi^-1(A)={x in R| phi(x) in A} is prime in R.b. If A is maximal in S, show that phi^-1(A) is maximal in R.

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