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

2.2. SUBMODULES AND FACTOR MODULES 23
(2) Let I ⊂ R be an ideal. Restriction of scalars along the projection R → R/I
gives any R/I-module as an R-module.
2.1.13. Proposition. Let R be a ring. There is a dictionary:
(1) To an R[X]-module N associate the pair (N, f) consisting of N as R-module
through restriction of scalars and f = XN : N → N, f(y) = Xy scalar
multiplication with X as an R-module homomorphism. An R[X]-homomorphism
g : N → N ′ gives an R-homomorphism such that g ◦ f = f ′ ◦ g.
(2) To a pair (N, f) of an R-module and a homomorphism f : N → N associate
the R[X]-module with abelian group N and scalar multiplication determined
by Xy = f(y) for y ∈ N. Note
( anX n )y = anf ◦n (y)
An R-homomorphism g : N → N ′ such that g ◦ f = f ′ ◦ g is an R[X]homomorphism.
Proof. The statement is an algorithm to follow.
2.1.14. Proposition. Let R be a ring and M a module. The abelian group R ⊕ M
with multiplication
(a + x)(b + y) = ab + (ay + bx)
is a ring. R is a subring and M is an ideal.
Proof. Simple calculations show that the conditions 1.1.2 are satisfied.
2.1.15. Exercise. (1) Show that a composition of homomorphisms is a homomorphism.
(2) Show that composition of scalar multiplications with a, b ∈ R on a module M is a
scalar multiplication with the product, aM ◦ bM = (ab)M .
(3) Let φ : R → S be a ring homomorphism. Show that φ is an R-module homomorphism,
when S is viewed as R-module through restriction of scalars 2.1.11.
(4) Fill out the dictionary 2.1.13.
2.2. Submodules and factor modules
2.2.1. Lemma. Let R be a ring and M a module. Let Nα be a family of submodules.
Then the additive subgroups
α Nα and
α Nα are submodules.
Proof. Use the formulas xα + yα = (xα + yα) and a xα = axα to
conclude that Nα is a submodule. If x, y ∈ Nα for all α, then x + y, ax ∈ Nα
for all α, so Nα is a submodule.
2.2.2. Definition. Let R be a ring and M a module. The intersection of all submodules
containing a subset Y ⊂ M is the submodule generated by Y and denoted
RY . This is the smallest submodule, 2.2.1, of M containing Y . The module M is
generated by Y if RY = M. Let I be an ideal. The submodule generated by all
products ax, a ∈ I, x ∈ M is denoted IM.
2.2.3. Proposition. Let R be a ring and M a module. If Y ⊂ M, then RY =
y∈Y Ry,
RY = {a1y1 + · · · + anyn|ai ∈ R, yi ∈ Y }
Proof. The righthand side is contained in the submodule RY . Moreover the righthand
side is a submodule containing Y , so equality.
2.2.4. Corollary. Let I ⊂ R be an ideal and M a module.