8.2. HOMOMORPHISMS AND QUOTIENT MODULES
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8.1.9. Let V be a finite dimensional vector space V over a field K . Let T 2 EndK.V /. Give V the corresponding KŒx–module structure defined by Pi ˛i xi v D Pi ˛i T i .v/. Show that V is not free as a KŒx–module.
8.1.10. Show that conditions (a) and (c) in Proposition 8.1.28 are equivalent.
8.2. Homomorphisms and quotient modules
In this section, we construct quotient modules and develop homomor phism theorems for modules, which are analogues of the homomorphism theorems for groups and rings. Recall that if M and N are R–modules, then Hom.M; N / denotes the set of R–module homomorphisms from M to N .
Proposition 8.2.1.
(a) If ' 2 HomR.M; N /, then ker.'/ is a submodule of M and '.M / is a submodule of N .
(b) If ' 2 HomR.M; N / and 2 HomR.N; P /, then ı ' 2 HomR.M; P /.
Proof. Exercise 8.2.1.
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Proposition 8.2.2.
(a) If ; ' 2 HomR.M; N /, define their sum by .' C /.m/ D '.m/C .m/. HomR.M; N / is an abelian group under addition.
(b) EndR.M / is a ring with addition defined as above and multiplication defined by composition.
Proof. Exercise 8.2.2.
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Let M be an R–module and N an R–submodule. We can form the quotient M=N as an abelian group and considier the quotient map W
M ! M=N as a homomorphism of abelian groups. In fact, M=N is an R–module and the quotient map is an R–module homomorphism.
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.
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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 m0 C N , then .m m0/ 2 N . Hence
rm
rm0 D r.m m0/ 2 N , since N is a submodule. But this means that rm C N D rm0 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: .r1r2/.m C N / D .r1r2/m C N D r1.r2m/ C N D r1.r2m C N / D r1.r2.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: .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.
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Example 8.2.4. If I is a left ideal in R, then R=I is an R–module with the action r .r1 C I / D rr1 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 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 ' ı D '. (See the following diagram.)
'
M
qqqqq
P
q qqqq
∼
= Q'
qqqq
M=N
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Proof. The homomorphism theorem for groups (Theorem 2.7.6) gives us an isomorphism of abelian groups Q ' W M=N ! P satisfying Q ' ı D '.
We have only to verify that Q ' also respects the R actions. But this follows at once from the definition of the R action on M=N : Q'.r.m C N // D Q'.rm C N / D '.rm/ D r'.m/ D r Q'.m C N /:
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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 an exercise.
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Proposition 8.2.8. Let ' W M ! M be a surjective R–module ho momorphism 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/.
Proof. Exercise 8.2.5.
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Proposition 8.2.9. (Factorization Theorem) Let ' W M ! M be a surjective 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
qqqqq
M
q qqqq
Q'
qqqq
M=N
Proof. Exercise 8.2.6.
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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 M . Then ' 1.'.A// D A C N D fa 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.
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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.
8.2.4. Complete the proof of the Correspondence Theorem, Proposition 8.2.7.
8.2.5. Prove Proposition 8.2.8.
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