College Algebra Exam Review 344

College Algebra Exam Review 344 - ±.rm D rm C N D r.m C N...

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354 8. MODULES Proposition 8.2.3. Let M be an R –module and N an R –submodule. Then the quotient M=N has the structure of an R –module and the quotient map ± W M ±! M=N is a homomorphism of R –modules. If R has identity and M is unital, then M=N is unital. Proof. We attempt to define the product of a ring element r and a coset m C N by the formula r.m C N/ D rm C N . As usual, when we define an operation in terms of representatives, we have to check that the operation is well defined. If m C N D m 0 C N , then .m ± m 0 / 2 N . Hence rm ± rm 0 D r.m ± m 0 / 2 N , since N is a submodule. But this means that rm C N D rm 0 C N , and the operation is well defined. Once we have checked that the action of R on M=N is well defined, it is easy to check that the axioms of an R –module are satisfied. For example, the mixed associative law is verified as follows: .r 1 r 2 /.m C N/ D .r 1 r 2 /m C N D r 1 .r 2 m/ C N D r 1 .r 2 m C N/ D r 1 .r 2 .m C N//: The quotient map ± W M ±! M=N is a homomorphism of abelian groups, and the definition of the R action on the quotient group implies that ± is an R –module homomorphism:
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Unformatted text preview: ±.rm/ D rm C N D r.m C N/ D r±.m/: The statement regarding unital modules is also immediate from the definition of the R –module structure on the quotient group. n Example 8.2.4. If I is a left ideal in R , then R=I is an R –module with the action r.r 1 C I/ D rr 1 C I . All of the homomorphism theorems for groups and rings have ana-logues for modules. Each of the theorems is proved by invoking the analo-gous theorem for abelian groups and then by checking that the homomor-phisms respect the R –actions. Theorem 8.2.5. (Homomorphism theorem for modules). Let ' W M ±! P be a surjective homomorphism of R –modules with kernel N . Let ± W M ±! M=N be the quotient homomorphism. There is an R –module isomorphism Q ' W M=N ±! P satisfying Q ' ı ± D ' . (See the following diagram.)...
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