# modules that are both injective projective

• January 5th 2012, 03:58 AM
ymar
modules that are both injective projective
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?
• January 5th 2012, 11:38 PM
NonCommAlg
Re: 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 $D_i.$