A unitary left Rmodule M
over a left noetherian ring R with identity is called admissible if every prime ideal of
R, which is the left annihilator of all nonzero submodules of a submodule N of
M, is also the left annihilator of all nonzero elements of N. The object of
this paper is to study left noetherian rings R whose category of unitary left
Rmodules consists of admissible modules. A ring R in this class of rings is
characterized by the fact that every injective left Rmodule is a direct sum of
injective and indecomposable modules of the form E_{R}(R∕P), where P denotes
a prime ideal of R and E_{R}(R∕P) the injective hull of the left Rmodule
R∕P.
Lesieur and Croisot have defined the heart C(E) of an injective module E
to be the intersection of the kernels of all endomorphisms in the Jacobson
radical of Hom_{R}(E,E), and the heart of any module M to be the submodule
C(M) = M ∩C(E(M)). Although the socle is always contained in the heart, the two
submodules are not equal in general, a simple example being a torsionfree abelian
group. This suggests the study of rings with the property that heart and socle
coincide in every one of their modules. In §3 the discussion is restricted to left
artinian rings, whose left modules are admissible, and it is shown that the class of
these rings is the class of all reduced left artinian rings, a ring being reduced if
it is a direct sum of division rings modulo its Jacobson radical. A ring R
in this class has the interesting property that for every prime ideal P the
heart of the left Rmodule R∕P is equal to its socle. But, although these
modules form an injective decomposition basis for the category of all unitary
left Rmodules, socle and heart can be different for some objects in this
category. In §4, however, it is shown that socle and heart coincide in every
module over a reduced left artinian ring R if and only if R is a direct sum of
finitely many local left artinian rings. This result admits two interesting
corollaries:
1. A commutative noetherian ring with identity is artinian if and only if heart and
socle are equal for every left Rmodule.
2. A left noetherian ring with identity whose modules are admissible
and have coinciding heart and socle has the ArtinRees property for left
ideals.
