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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 denition 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 analogues for modules. Each of the theorems is proved by invoking the analogous theorem for abelian groups and then by checking that the homomorphisms 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 ' D ' . (See the following diagram.)...
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 Fall '08
 EVERAGE
 Algebra

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