Unformatted text preview: (a) Let Q R D Z ´ R , as an abelian group. Give Q R the multiplication .n;r/.m;s/ D .nm;ns C mr C rs/: Show that this makes Q R into a ring with multiplicative identity .1;0/ . (b) Show that r 7! .0;r/ is a ring isomorphism of R into Q R with image f g ´ R . Show that f g ´ R is an ideal in Q R . (c) Show that if ' W R µ! S is a homomorphism of R into a ring S with multiplicative identity 1, then there is a unique homomorphism Q ' W Q R µ! S such that Q '..0;r// D '.r/ and Q '..1;0// D 1 . 6.3. Quotient Rings In Section 2.7 , it was shown that given a group G and a normal subgroup N , we can construct a quotient group G=N and a natural homomorphism from G onto G=N . The program of Section 2.7 can be carried out more or less verbatim with rings and ideals in place of groups and normal subgroups:...
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 Fall '08
 EVERAGE
 Algebra, Group Theory, Normal subgroup, Abelian group, Za C Ra C, Za C Ra

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