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A unitary left R-module 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
R-modules consists of admissible modules. A ring R in this class of rings is
characterized by the fact that every injective left R-module is a direct sum of
injective and indecomposable modules of the form ER(R∕P), where P denotes
a prime ideal of R and ER(R∕P) the injective hull of the left R-module
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 HomR(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 torsion-free 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 R-module R∕P is equal to its socle. But, although these
modules form an injective decomposition basis for the category of all unitary
left R-modules, 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 R-module.
2. A left noetherian ring with identity whose modules are admissible
and have coinciding heart and socle has the Artin-Rees property for left
ideals.
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