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College Algebra Exam Review 346

College Algebra Exam Review 346 - M Then ± 1.A D A C N D f...

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356 8. MODULES Proof. Exercise 8.2.5 . n Proposition 8.2.9. (Factorization Theorem) Let ' W M ! M be a sur- jective homomorphism of R –modules with kernel K . Let N K be a submodule , and let W M ! M=N denote the quotient map. Then there is a surjective homomorphism Q ' W M=N ! M such that Q ' ı D ' . (See the following diagram.) The kernel of Q ' is K=N M=N . M ' q q M q q q q Q ' M=N Proof. Exercise 8.2.6 . n Proposition 8.2.10. (Diamond Isomorphism Theorem) Let ' W M ! M be a surjective homomorphism of R –modules with kernel N . Let A be a submodule of
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Unformatted text preview: M . Then ' ± 1 .'.A// D A C N D f a C n W a 2 A and n 2 N g : Moreover, A C N is a submodule of M containing N , and .A C N/=N Š '.A/ Š A=.A \ N/: Proof. Exercise 8.2.7 . n Exercises 8.2 R denotes a ring and M an R –module. 8.2.1. Prove Proposition 8.2.1 . 8.2.2. Prove Proposition 8.2.2 . 8.2.3. Let I be an ideal of R . Show that the quotient module M=IM has the structure of an R=I –module....
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