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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.

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. Von Neumann regularity is also equivalent to all right modules being flat, and is also equivalent to all right modules being principally injective (a.k.a. "divisible" in Lam's *Lectures on modules and rings.")

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 modules is injective if and only if the ring is Noetherian, (Lam 1999, Th 3.46).

Classically semisimple: every module has complemented lattice of submodules.

Coherent: Chase theorem

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. )

Classically semisimple: every module has complemented lattice of submodules.

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 injective indecomposables.

Classically semisimple: every module ahs complemented lattice of submodules.

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.

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.

Classically semisimple: every module has complemented lattice of submodules.

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)