College Algebra Exam Review 272

# College Algebra Exam Review 272 - (a Let Q R D Z ´ R as an...

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282 6. RINGS 6.2.16. Show that a nonzero homomorphism of a simple ring is injective. In particular, a nonzero homomorphism of a field is injective. 6.2.17. Show that the intersection of any family of ideals in a ring is an ideal. Show that the ideal generated by a subset S of a ring R is the inter- section of all ideals J of R such that S J R . 6.2.18. Let I and J be two ideals in a ring R . Show that I C J D f a C b W a 2 I and b 2 J g is an ideal in R . 6.2.19. Let I and J be two ideals in a ring R . Show that IJ D f a 1 b 1 C a 2 b 2 C C a s b s W s 1; a i 2 I; b i 2 J g is an ideal in R , and IJ I \ J . 6.2.20. Let R be a ring without identity and a 2 R . Show that the ideal generated by a in R is equal to Z a C Ra C aR C RaR , where Z a is the abelian subgroup generated by a , Ra D f ra W r 2 R g , and so on. Show that if R is commutative, then the ideal generated by a is Z a C Ra . 6.2.21. Let M be an ideal in a ring R with identity, and a 2 R n M . Show that M C RaR is the ideal generated by M and a . How must this statement be altered if R does not have an identity? 6.2.22. Let R be a ring without identity. This exercise shows how R can be imbedded as an ideal in a ring with identity.
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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 ho-momorphism 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 sub-groups:...
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