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

2.3. KERNEL AND COKERNEL 27
Proof. Let f(a) = ax. f(ab) = (ab)x = a(bx) = af(b) shows that f is a
homomorphism. This argument does not use the commutativity 1.1.2 of R.
2.3.11. Definition. Denote the homomorphism 2.3.10
The annihilator of x is the ideal
1x : R → M, a ↦→ ax
Ann(x) = Ker 1x = {a ∈ R|ax = 0}
For a subset Y ⊂ M the annihilator Ann(Y ) =
y∈Y Ann(y) is the ideal of
elements
Ann(Y ) = {a ∈ R|ay = 0, for all y ∈ Y }
2.3.12. Proposition. Let R be a ring and Y a subset of a module M.
(1) Ann(Y ) = Ann(RY ).
(2) a ∈ Ann(M) if and only if aM = 0.
(3) If modules M M ′ then Ann(M) = Ann(M ′ ).
(4) The induced homomorphism
is an isomorphism.
1 ′ x : R/ Ann(x) → Rx, a + Ann(x) ↦→ ax
Proof. (1) If a ∈ Ann(Y ) then a biyi = biayi = 0 giving the not so obvious
Ann(Y ) ⊂ Ann(RY ). (2) Clear since aM(x) = ax, 2.1.7. (3) Let f : M → M ′
be an isomorphism and a ∈ R. Then af(x) = f(ax) expresses that f ◦ aM =
aM ′ ◦ f. Since f is bijective, aM = 0 if and only if aM ′ = 0. By (2) Ann(M) =
Ann(M ′ ). (4) This follows from 2.3.5.
2.3.13. Corollary. Let I, J ⊂ R be ideals.
(1) Ann(R/I) = I
(2) If R/I R/J then I = J.
2.3.14. Lemma. Let I ⊂ R be an ideal and M an R-module. If I ⊂ Ann(M)
then M is an R/I-module with the scalar multiplication
That is, M is an R/ Ann(M)-module.
R/I × M → M, (a + I, x) ↦→ ax
2.3.15. Example. Let a ∈ R give scalar multiplication aM : M → M, x ↦→ ax.
(1) a ∈ Ann(Ker aM).
(2) a ∈ Ann(Cok aM).
(3) Ker aM and Cok aM are modules over the factor ring R/(a), 2.3.14.
2.3.16. Definition. Let R be a ring and L, N ⊂ M submodules. The colon ideal
N : L is the ideal of elements a ∈ R such that aL ⊂ N.
2.3.17. Proposition. The colon ideal of submodules L, N ⊂ M is
N : L = Ann(N + L/N)
Proof. If a ∈ N : L then aL ⊂ N. Therefore a(N + L) ⊂ N and a ∈ Ann(N +
L/N). Conversely if a ∈ Ann(N + L/N) then aL ⊂ N and therefore a ∈ N :
L.
2.3.18. Exercise. (1) Give an example of a homomorphism f : M → N submodules
M1, M2 ⊂ M such that f(M1 ∩ M2) = f(M1) ∩ f(M2).