Equivalence of Definitions of Projective Module

Theorem
Let $A$ be a ring.

Let $M$ be an $A$-module.

The following are equivalent:


 * (1) $M$ is a projective module, i.e. $M$ is a projective object in the category of left $A$-modules.
 * (2) $M$ is a direct summand of a free module.
 * (3) Every short exact sequence of the form

\begin{align*} \xymatrix{ 0 \ar[r] & X \ar[r]^f & Y \ar[r]^g & M \ar[r] & 0 } \end{align*} splits, i.e. there is a homomorphism $s : M \to Y$ with $g \circ s = \mathrm{id}_M$.

(1) implies (2)
By Surjection by Free Module there is a free module $Y$ and a surjection $g : Y \to M$.

By Epimorphism of modules iff surjective $g$ is an epimorphism.

By Definition:Projective Object applied to $\mathrm{id}_M$, there is a homomorphism $s : M \to Y$ with $g \circ s = \mathrm{id}_M$.

We have $Y = \mathrm{im}(s) \oplus \ker(g)$.

(2) implies (1)
Assume, that there is an $A$-module $Q$, such that $M \oplus Q$ is free.

Let $f : Y \to Z$ be an epimorphism.

Let $h : M \to Z$ be a homomorphism.

Let $S$ be a basis of $M \oplus Q$.

By Epimorphism of modules iff surjective $f$ is surjective.

Hence for all $s \in S$, there is some $y_s \in Y$ with $f(y_s) = h(\mathrm{pr}_1(s))$, where $\mathrm{pr}_1 : M \oplus Q \to M$ denotes projection to $M$.

By Universal Property of Free Modules there is a unique homomorphism $t : M \oplus Q \to Y$, such that $f \circ t = h \circ \mathrm{pr}_1$.

We have $f \circ (t \circ i_1) = h \circ \mathrm{pr}_1 \circ i_1 = h$, where $i_1 : M \to M \oplus Q$ is the inclusion of the first summand.

It follows, that $t \circ i_1$ is the desired lift of $f$.

(3) implies (2)
By Surjection by Free Module there is a free module $Y$ and a surjection $g : Y \to M$.

There is a short exact sequence \begin{align*} \xymatrix{ 0 \ar[r] & \ker(g) \ar[r] & Y \ar[r]^g & M \ar[r] & 0 } \end{align*}

By Structure of Split Exact Sequence there is an isomorphism $Y \cong M \oplus \ker(g)$.