Projective modules were introduced in 1956 by Cartan and Eilenberg in their book Homological Algebra. Does anyone know why they chose the word "projective"? Does it have something to do with the notion of projection?
The term "projection" has a few possible meanings in linear algebra, and they are equivalent to the property of being a projective module.
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.