College Algebra Exam Review 264

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

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern