Vector spaces are both projective and injective if the axiom of choice holds. Are there any other modules with this property? Do they have a good characterization?
there's a characterization of rings over which every module is both projective and injective. these rings are called semisimple and they are in the form $\displaystyle \bigoplus_{i=1}^k M_{n_i}(D_i),$ where $\displaystyle k \geq 1$ is any integer, each $\displaystyle D_i$ is a division ring and $\displaystyle M_{n_i}(D_i)$ is the ring of $\displaystyle n_i \times n_i$ matrices with entries in $\displaystyle D_i.$