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Let p: R → S be a ring homomorphism and let PC S be a prime ideal of S. Consider the set I := {a €R| y(a) e P}. Which of the following statements are true? Select all that apply: O For any R, S, p, and P, the set I is a prime ideal of R. O For some R, S, p, and P, the set I is an ideal but not a prime ideal of R. O For some R, S, p, and P, the set I is a maximal ideal of R. O For any R, S, 9, and P, the set I is an ideal of R. O For some R, S, p, and P, the set I is not an ideal of R.

Let p: R → S be a ring homomorphism and let PC S be a prime ideal of S. Consider the set I := {a €R| y(a) e P}. Which of the following statements are true? Select all that apply: O For any R, S, p, and P, the set I is a prime ideal of R. O For some R, S, p, and P, the set I is an ideal but not a prime ideal of R. O For some R, S, p, and P, the set I is a maximal ideal of R. O For any R, S, 9, and P, the set I is an ideal of R. O For some R, S, p, and P, the set I is not an ideal of R.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.4: Maximal Ideals (optional)
Problem 13E
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Let p: R → S be a ring homomorphism and let PC S be a prime ideal of S. Consider the set I := {a €R| y(a) e P}. Which of the following statements are true? Select all that apply: O For any R, S, p, and P, the set I is a prime ideal of R. O For some R, S, p, and P, the set I is an ideal but not a prime ideal of R. O For some R, S, p, and P, the set I is a maximal ideal of R. O For any R, S, 9, and P, the set I is an ideal of R. O For some R, S, p, and P, the set I is not an ideal of R.
Transcribed Image Text:Let p: R → S be a ring homomorphism and let PC S be a prime ideal of S. Consider the set I := {a €R| y(a) e P}. Which of the following statements are true? Select all that apply: O For any R, S, p, and P, the set I is a prime ideal of R. O For some R, S, p, and P, the set I is an ideal but not a prime ideal of R. O For some R, S, p, and P, the set I is a maximal ideal of R. O For any R, S, 9, and P, the set I is an ideal of R. O For some R, S, p, and P, the set I is not an ideal of R.
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