Origin of the terminology projective module

Solution 1:

The term "projection" has a few possible meanings in linear algebra, and they are equivalent to the property of being a projective module.

1. A linear operator $f \colon {\mathbf R}^n \rightarrow {\mathbf R}^n$ is geometrically a projection to a subspace in some coordinate system if and only if $f^2 = f$.

2. For two vector spaces $V$ and $W$, the function $V \oplus W \rightarrow V$ where $(v,w) \mapsto v$ is called a projection (out of the direct sum).

For a commutative ring $R$ and $R$-module $P$, the following properties that abstract the above two conditions are both equivalent to $P$ being a projective module.

1. There is a free $R$-module $F$ and an $R$-linear map $f \colon F \rightarrow F$ such that $f^2 = f$ and $f(F) \cong P$ (so $P$ is isomorphic to the image of a "projection").

2. Any way of making $P$ look like a quotient module essentially amounts to making it the image of the natural projection out of a direct sum module to one of its direct summands: if $f \colon M \twoheadrightarrow P$ is any surjective $R$-linear map then there is an $R$-module isomorphism $h \colon M \cong P \oplus K$ for some $R$-module $K$ such that $h(m) = (f(m),*)$ for all $m \in M$ (thus $f$ looks like the natural projection map $P \oplus K \rightarrow P$).