modules that are both injective projective

Quote:

Originally Posted by ymar

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 $\bigoplus_{i=1}^k M_{n_i}(D_i),$ where $k \geq 1$ is any integer, each $D_i$ is a division ring and $M_{n_i}(D_i)$ is the ring of $n_i \times n_i$ matrices with entries in $D_i.$