Abstract: An n x n-matrix P is called a projection matrix if P is an idempotent, i.e., P^2 = P; in this context the notion seems anodyne. If our matrices have coefficients in a more complicated algebraic structure, say a commutative ring R, then the image of a projection matrix is called a projective module. I would like to explain the importance of social factors in the emergence of projective modules as an object of mainstream mathematical focus in the mid to late 1950s. My main aim is to argue that "mainstream focus'' in mathematics is socially directed. Along the way, I aim to highlight what one might call a disjunction between the reasons we use to explain to students why we make definitions and why we make particular choices in research mathematics. I also aim to reflect on the question: what do mathematicians want from the history of mathematics?