Topics in algebra Chapter IV: Commutative rings and modules I - 1

5 Projective modules and injective modules
We first introduce the notion: free modules.
Definition 5.1 Let F be an R-module. F is a free module if there is a
subset X = {ei | i ∈ Λ} (which is called a basis) of F such that F = (X),
i.e., for every x ∈ F , there are e1, . . . , en ∈ X and r1, . . . , rn ∈ R such
n
that x = riei, and that X is linear independent, i.e., for every finite set
i=1
{e1, . . . , en} of X with
F ∼ = ⊕i∈ΛR.
n
riei = 0, then ri = 0 for every i. In this case,
i=1
If F is free with a finite basis then F ∼ = R n for some n and F is called a free
R-module of rank n.
We now introduce the notion: projective modules.
Definition 5.2 Let P be an R-module. P is said to be projective if for any
exact sequence M g
−→ N −→ 0 and R-homomorphism f : P −→ N there is
an R-homomorphism h : P −→ M such that g ◦ h = f.
It is easy to see that free modules are projective.
There are several equivalent characterizations for projective modules.
Proposition 5.3 Let R be a ring. Then the following are equivalent for an
R-module P :
(a) P is projective.
(b) Any s.e.s. of the form 0 −→ L f
−→ M g
−→ P −→ 0 splits.
(c) P is a direct summand of some free R-module F .
Proof. (a) ⇒ (b) Let 0 −→ L f
−→ M g
−→ P −→ 0 be a s.e.s.. Let i : P −→ P
be the identity map; then there is a R-homomorphism h : P −→ M such
that g ◦ h = i, so that g ◦ h = idP ,i.e., the s.e.s. splits.
(b) ⇒ (c) Let X = {xi | i ∈ Λ} be a generating set of P ; then there is a free
module F = ⊕i∈ΛR and a R-epimorphism g : F −→ P . Let K be the kernel
of g; then 0 −→ K i
−→ F g
−→ P −→ 0 is a s.e.s., so that the sequence splits
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