TITLE: Properties of rings that have an elegant description in terms of the associated category of modules
QUESTION [12 upvotes]: Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant description in terms of the associated (concrete) category of modules?

REPLY [4 votes]: If $R$ is an integral domain, then every ideal factors into primes if and only if every submodule of a projective module is projective.