College Algebra Exam Review 264

# College Algebra Exam Review 264 - 274 Example 6.2.10 The...

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274 6. RINGS Example 6.2.10. The map P i k i x i 7! P i OEk i Łx i is a homomorphism of Z OExŁ to Z n OExŁ . Example 6.2.11. Let R be a commutative ring with multiplicative iden- tity element. Then ROEx; yŁ Š ROExŁOEyŁ . To prove this, we use the one– and two–variable substitution principles to produce homomorphisms from ROEx; yŁ to ROExŁOEyŁ and from ROExŁOEyŁ to ROEx; yŁ . We have injective homomorphisms ' 1 W R ! ROExŁ and ' 2 W ROExŁ ! ROExŁOEyŁ . The composition ' D ' 2 ı ' 1 is an injective homomorphism from R into ROExŁŁyŁ . By the two variable substitution principle, there is a unique homomorphism ˚ W ROEx; yŁ ! ROExŁOEyŁ which extends ' and sends x 7! x and y 7! y . Now we produce a map in the other direction. We have an injective ho- momorphism W R ! ROEx; yŁ . Applying the one variable substitution principle once gives a homomorphism 1 W ROExŁ ! ROEx; yŁ extending and sending x 7! x . Applying the one variable substitution principle a second time gives a homomorphism W ROExŁOEyŁ ! ROEx; yŁ extending 1
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