From http://en.wikipedia.org/wiki/Injective_module
Every submodule of every injective module is injective if and only if the ring is Artinian semisimple (Golan & Head 1991, p. 152); every factor module of every injective module is injective if and only if the ring is hereditary, (Lam 1999, Th. 3.22).
Every infinite direct sum of injective right modules is injective if and only if the ring is right Noetherian, (Lam 1999, Th 3.46). Every infinite direct product of flat right modules is flat iff the ring is left coherent, and every infinite direct product of projective right modules is projective iff the ring is right perfect and left coherent. (The latter two theorems appear in this paper by Stephen Chase. )
Another characterization of (one sided, as above) noetherianity with injective modules: every injective is direct sum of indecomposables; there is only a set (not proper class) of isomorphism types of injective indecomposables.
Dually one has many characterization results concerning projectivity; for example for a commutative integral domain, one of the many equivalent conditions to be Dedekind (Pr"ufer) is hereditary (semi-heredirary).
QF rings: every projective is injective (on one side) iff every injective is projective (on one side) iff the ring is two-sided artinian and the finitely generated right and left modules are in natural duality (with the usual dual, hom to the ring of scalars) iff the ring is (one sided) noetherian and in the Galois correspondence "annihilator" between right and left ideals, each element is closed.
Principal ideal artinian rings: every homomorphic image is QF iff finite direct products of matrix rings over CPU rings (rings $R$ with a nilpotent element $p$ such that $pR=Rp$ and the $p^nR=Rp^n$ are all one sided ideals) iff each finitely generated module has a lattice of submodules which is finite direct product of primary lattices. [Generalizing primary decomposable to semiprimary characterizes artinian serial rings]
von Neumann regular: $\forall a\exists x:axa=a$ iff each in each finitely presented (or even only finitely generated projective) module the semilattice of finitely generated submodules is a complemented sublattice of the lattice of all submodules.
Classically semisimple: every module has complemented lattice of submodules.
One can multiply examples, ad enjoy finding them in the books by Lam, Rowen, Faith, Stenstr"om, ...
However: all the magic disappares when one notes that every Morita invariant property can be defined by a purely categorical property of the abelian category of all right modules, and when in such a category one fixes a progenerator then any property of the ring can be expressed in this language (compare semisimple rings, a Morita invariant property, versus skew fields, characterized by freenes of modules instead of injectivity / projectivity). Besides, all these characterizations are also possible in lattice theoretic terms (Hutchinson - Isbell lattice associated to a abelian category); when the property to be expressed is not Morita - invariant one needs to fix a basis in the lattice.
So the problem is not the possibility to express a ring property in the categorical language of modules (bicartesian language of abelian categories, or the language of lattice theory). The problem is to formalize the requested "elegance" of the characterization, a problem with no clear mathematical answer. Not a true formalization, but a guide might be: seach for equivalence between properties valid for all modules (or somewhat "unbounded" classes of modules: all finitely generated modules; all projetive modules; ...) and properties that can be expressed "internally" in the ring (or for example in its $2\times 2$ matrix ring, or that in any case depend on a class of modules that is "bounded", like $n$-generated modules for a fixed $n$. Example: a ring is unit regular iff the ring is vNr and perspectivity is transitive in the lattice of the $2\times 2$ matrix ring iff the ring is vNr and perpectivity is transitive in all the lattices of finitely generated submodules of a finitely presented module iff the ring is vNr and cancellation is valid in the additive monoid of isomorphism types of finitely presented modules)
