Vol. 33, No. 2, 1970

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Admissible modules and a characterization of reduced left artinian rings

Günter Krause

Vol. 33 (1970), No. 2, 291–309

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.

Mathematical Subject Classification
Primary: 16.50
Received: 23 December 1968
Published: 1 May 1970
Günter Krause