**Exercise: Properties of Ring Homomorphisms****3. Let $ f : R \to S $ be a ring homomorphism**(a) If $ I $ is an ideal of $ S $, prove the pre-image of $ I $, that is $ f^{-1}(I) $, is an ideal of $ R $.(b) If $ P $ is a prime ideal of $ S $, prove $ f^{-1}(P) $ is a prime ideal of $ R $.(c) If $ f $ is onto, prove that if $ J $ is any ideal of $ S $, then $ J = f(I) $ where $ I $ is an ideal of $ R $ containing $ \text{ker} f $.

Let I be an ideal of a ring R , and Let f:R→S be a ring homomorphism. To prove: f-1I is an ideal of…

Answered: 3. Let f: RS be a ring homomorphism (a)…

3. Let f: RS be a ring homomorphism (a) If I is an ideal of S, prove the pre-image of I, that is f-¹(I), is an ideal of R

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Exercise: Properties of Ring Homomorphisms**

**3. Let $ f : R \to S $ be a ring homomorphism**

(a) If $ I $ is an ideal of $ S $, prove the pre-image of $ I $, that is $ f^{-1}(I) $, is an ideal of $ R $.

(b) If $ P $ is a prime ideal of $ S $, prove $ f^{-1}(P) $ is a prime ideal of $ R $.

(c) If $ f $ is onto, prove that if $ J $ is any ideal of $ S $, then $ J = f(I) $ where $ I $ is an ideal of $ R $ containing $ \text{ker} f $.

Expert Solution

Step 1

Let I be an ideal of a ring R , and Let $f : R \to S$ be a ring homomorphism.

To prove: $f^{- 1} (I)$ is an ideal of R.

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