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.

Brittney Lord 2020-11-22 Answered

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 ϕ1(A)={xRϕ(x)A} is prime R.
b. If A is maximal in S, show that ϕ1(A) is maximal R.

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Expert Answer

stuth1
Answered 2020-11-23 Author has 97 answers

a. Suppose A i sprime ideal in S.
Consider abϕ1(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 ϕ1(A)
Thus, ϕ1(A) is prime R
b. Consider the homomorphism phi: RSA defined by ϕ(r)=ϕ(r)+A
Then the kernel of ϕ is ϕ1(A).
Then,RkerϕSA
Therefore, ϕ1(A) is maximal ideal.

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