Commutative algebra - Department of Mathematical Sciences - old ...

26 2. MODULES
2.3.6. Corollary. Let p : M → M/N be the projection onto a factor module. The
map L ′ ↦→ L = p −1 (L ′ ) gives a bijective correspondence between submodules
in M/N and submodules in M containing N. Also L ′ = p(L) = L/N. This
correspondence preserves inclusions, additions and intersections of submodules.
Proof. If L ′ is a submodule of M/N then clearly p(p−1 (L ′ )) = L ′ . If N ⊂ L
is a submodule of M then clearly L ⊂ p−1 (p(L)). Moreover if x ∈ p−1 (p(L))
then p(x) = p(y) for some y ∈ L and therefore y − x ∈ N. It follows that
L = p−1 (p(L)) and the correspondence is bijective. Inclusions are easily seen to
be preserved. Also easily p(L1 + L2) = p(L1) + p(L2) and p−1 (L ′ 1 ∩ L′ 2 ) =
p−1 (L ′ 1 ) ∩ p−1 (L ′ 2 ) hold. The two resting equalities are consequences of this and
bijectivity of the correspondence.
2.3.7. Corollary. Let L ⊂ N ⊂ M be submodules. Then there is a canonical
isomorphism
M/N → (M/L)/(N/L)
Proof. The kernel of the surjective east-south composite
M
M/N
M/L
(M/L)/(N/L)
is N. By 2.3.5 the horizontal lower factor map gives the isomorphism.
2.3.8. Corollary. Let L, N ⊂ M be submodules. Then there is a canonical isomorphism
N/N ∩ L → N + L/L
given by x + N ∩ L ↦→ x + L.
Proof. The kernel of the east-south composite
N
N/N ∩ L
N + L
N + L/L
is N ∩ L. Since x + y + L = x + L for x ∈ N, y ∈ L this composite is also
surjective. By 2.3.5 the horizontal lower factor map gives the isomorphism.
2.3.9. Proposition. Let f : M → N and g : N → L be homomorphisms such that
Im f ⊂ Ker g. Then there is a unique homomorphism g ′ : Cok f → L such that
g = g ′ ◦ p.
Proof. This follows from 2.3.5.
M f
N
g
p
L
g ′
Cok f
2.3.10. Lemma. Let R be a ring and M a module. For x ∈ M the map R →
M, a ↦→ ax is the unique homomorphism such that 1 ↦→ x.