On the endomorphism ring of a module with relative chain conditions

A well-known result of Small states that if M is a noetherian left R-module having endomorphism ring S then any nil subring of S is nilpotent. Fisher [4] dualized this result and showed that if M is left artinian then any nil ideal of S is nilpotent. He gave a bound on the indices of nilpotency of nil subrings of the endomorphism rings of noetherian modules and raised the dual question of whether there are such bounds in the case of artinian modules. He gave an affirmative answer if the module is also assumed to be finitely-generated. Similar affirmative answers for modules with finite homogeneous length were given in [10] and [15]. On the other hand, the nilpotence of certain ideals of the endomorphism rings of modules noetherian relative to a torsion theory has been extensively studied. See [2,6,8,12,15,17]. Jirasko [11] dualized, in some sense, some of the results of [6] to torsion modules satisfying the descending chain conditions with respect to some radical.

In this paper we give a bound of indices of nilpotency on nil subrings of the endomorphism ring of a left R-module which is T-torsionfree with respect to some torsion theory T on R-mod. As a special case, we obtain an affirmative answer to Fisher's question. We also note that our results can be stated in an arbitrary Grothendieck category.     

The endomorphism ring of a Δ-module over a right noetherian ring

LetR be a right noetherian ring. A moduleM _R is called a Δ-module providedR satisfies the descending chain condition for annihilators of subsets ofM. For a Δ-module, a series 0?M ₁?M ₂?...?M _n =M can be constructed in which the factorsM _i /M _i?1 are sums of, α _i -semicritical modules where α₁≦α₂≦...≦α _n . In this paper we utilize this series in studying Λ=End(M _R ). It is shown that ifN={f∈Λ|Kerf is essential inM}, thenN is nilpotent. Specific bounds on the index of nilpotency are given in terms of this series. Further ifM is injective and α-smooth, the annihilators of the factors of this series are used to provide necessary and sufficient conditions for EndM _R to be semisimple.     

Modules semicocritical with respect to a torsion theory and their applications

The concepts of primitive ideal and semicocritical module with respect to a torsion theory are studied and related to the structure of torsionfree injective modules. Applications are made to the study of (1) composition series with respect to a torsion theory and (2) the structure of endomorphism rings of torsionfree modules. These results are natural generalizations of the properties of certain modules over (noetherian) rings with Krull dimension.     

On modules whose endomorphism ring is local

The paper is a study of modules with a local endomorphism ring. We consider decompositions of modules and give a simple proof of Azumaya’s theorem. We also define an equivalence relation on the family of direct summands of a module, and show that the properties of a decomposition are closely related to the properties of this equivalence relation. Supported by the Norwegian Research Council for Science and the Humanities.     

Endomorphism rings of essential extensions of a noetherian module

Nil subrings of the ring of endomorphisms of the rational completion of a noetherian module are nilpotent. If the quasi-injective hull of a noetherian module is contained in its rational completion, then the ring of endomorphisms of the former is semi-primary.     

Vanishing of stable homology with respect to a semidualizing module

We investigate stable homology of modules over a commutative noetherian ring R with respect to a semidualzing module C, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of the theory forces C to be trivial and R to be Gorenstein.     

Ratliff-Rush closures of ideals with respect to a Noetherian module

Let R be a commutative Noetherian ring, E a non-zero finitely generated R-module and I a E-proper ideal of R. The purpose of this paper is to provide some new characterizations of when all powers of I are Ratliff-Rush closed with respect to E and to answer a question raised by W. Heinzer et al. in (The Ratliff-Rush Ideals in a Noetherian Ring: A Survey, in Methods in Module Theory, Dekker, New York, 1992, pp. 149-159).