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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- Fall '08
- Algebra, Normal subgroup, Homomorphism, Abelian, isomorphism theorem