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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

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
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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**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 \( \text{ker} f \).
Transcribed Image Text:**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 \( \text{ker} f \).
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Step 1

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 R.

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