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Unformatted text preview: 8.2. HOMOMORPHISMS AND QUOTIENT MODULES ' M qqqqq
qqqqq 355 P qqqqqqqqqq
q ∼ ='
Q qqqqq qqqqq
Proof. The homomorphism theorem for groups (Theorem 2.7.6) gives us
an isomorphism of abelian groups ' W M=N ! P satisfying ' ı D ' .
We have only to verify that ' also respects the R actions. But this follows
at once from the deﬁnition of the R action on M=N :
'.r.m C N // D '.rm C N / D '.rm/
D r'.m/ D r '.m C N /:
Example 8.2.6. Let R be any ring, M any R–module, and x 2 R. Consider the cyclic R–submodule Rx . Then r 7! rx is an R–module homomorphism of R onto Rx . The kernel of this map is called the annihilator
of x ,
ann.x/ D fr 2 R W rx D 0g:
Note that ann.x/ is a submodule of R, that is a left ideal. By the homomorphism theorem, R=ann.x/ Š Rx .
Proposition 8.2.7. (Correspondence Theorem) Let ' W M ! M be an
R–module homomorphism of M onto M , and let N denote its kernel.
Then A 7! ' 1 .A/ is a bijection between R–submodules of M and R–
submodules of M containing N .
Proof. By Proposition 2.7.12, A 7! ' 1 .A/ is a bijection between the
subgroups of M and the subgroups of M containing N . It remains to
check that this bijection carries submodules to submodules. This is left as
I Proposition 8.2.8. Let ' W M ! M be a surjective R–module homomorphism with kernel K . Let N be a submodule of M and let N D
' 1 .N /. Then m C N 7! '.m/ C N is an isomorphism of M=N onto
M =N . Equivalently, M=N Š .M=K/=.N=K/. ...
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- Fall '08