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College Algebra Exam Review 273

# College Algebra Exam Review 273 - 6.3 QUOTIENT RINGS 283...

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Unformatted text preview: 6.3. QUOTIENT RINGS 283 For a ring R and an ideal I , we can form the quotient group R=I , whose elements are cosets a C I of I in R . The additive group operation in R=I is .a C I/ C .b C I/ D .a C b/ C I . Now attempt to define a multiplication in R=I in the obvious way: .a C I/.b C I/ D .ab C I/ . We have to check that this this is well defined. But this follows from the closure of I under multiplication by elements of R ; namely, if a C I D a C I and b C I D b C I , then .ab a b / D a.b b / C .a a /b 2 aI C Ib I: Thus, ab C I D a b C I , and the multiplication in R=I is well defined. Theorem 6.3.1. If I is an ideal in a ring R , then R=I has the structure of a ring, and the quotient map a 7! a C I is a surjective ring homomorphism from R to R=I with kernel equal to I . If R has a multiplicative identity, then so does R=I , and the quotient map is unital. Proof. Once we have checked that the multiplication in R=I is well de- fined, it is straightforward to check the ring axioms. Let us include onefined, it is straightforward to check the ring axioms....
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