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Unformatted text preview: 274 6. RINGS Example 6.2.10. The map P i k i x i 7! P i OEk i x i is a homomorphism of Z OEx to Z n OEx . Example 6.2.11. Let R be a commutative ring with multiplicative iden- tity element. Then ROEx;y ROExOEy . To prove this, we use the one and twovariable substitution principles to produce homomorphisms from ROEx;y to ROExOEy and from ROExOEy to ROEx;y . We have injective homomorphisms ' 1 W R ! ROEx and ' 2 W ROEx ! ROExOEy . The composition ' D ' 2 ' 1 is an injective homomorphism from R into ROExy . By the two variable substitution principle, there is a unique homomorphism W ROEx;y ! ROExOEy 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;y . Applying the one variable substitution principle once gives a homomorphism 1 W ROEx ! ROEx;y extending and sending x 7! x . Applying the one variable substitution principle a second time gives a homomorphism...
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- Fall '08