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Prove that if Rand S are commutative rings, and R "" S, then R[x] "" S[x]

Chapter 34, Problem 34.12

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QUESTION:

Prove that if R and S are commutative rings, and \(R \approx S\), then \(R[x] \approx S[x]\).

Text Transcription:

R approx S

R[x] approx S[x]

Questions & Answers

QUESTION:

Prove that if R and S are commutative rings, and \(R \approx S\), then \(R[x] \approx S[x]\).

Text Transcription:

R approx S

R[x] approx S[x]

ANSWER:

Step 1 of 6

Now we will show that F is injective:

Let  satisfies  

Now  which implies that  and then

And since f is injective we deduce, so that  

Thus,  

Therefore, F is injective

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