  It is proved that, for a left hereditary ring, an arbitrary left module has a
representation in the form of the direct sum of a stable left module and
indecomposable projective left modules (if and only if an arbitrary left module
has a representation in the form of the direct sum of a stable left module and
a projective left module) if and only if the ring is left perfect and right
coherent. In that case, the above-mentioned representations are unique up to
isomorphism; the latter representation is also functorial. The essential
ingredient in the proofs of the above-mentioned statements is a certain purely
categorical result. These statements, in particular, imply that, for any
principal ideal domain that is not a field, the fundamental theorem on finitely
generated modules over it can not be generalized to the case of all modules.
Moreover, with the aid of the above-mentioned categorical approach, we give a
new proof of the Zheng-Xu He's result asserting that any module of a ring has a
unique up to isomorphism injective-costable decomposition if and only if the
ring is left hereditary and left Noetherian. The above-mentioned statements, in
particular, imply that if the category of left modules over a left hereditary
ring is Krull-Schmidt, then the ring is left Artinian. Yet another criterion
for a ring to be left hereditary, left perfect and right coherent (resp. left
hereditary left Noetherian) found in the paper requires that the pair $(Stable$
$modules$, $Projective$ $modules)$ (resp. $(Injective$ $Modules, Costable$ $
Modules)$) of module classes be a pre-torsion theory. This implies that the
pair $(Stable$ $modules$, $Projective$ $modules)$ is a torsion theory if and
only if the ring is left hereditary and the injective envelope of the ring,
viewed as a left module over itself, is projective.

Rings and Algebras;Category Theory