Modules with Semi-Local Endomorphism Ring

On Endomorphism Rings and Dimensions of Local Cohomology Modules

Let (R, m) denote an n-dimensional complete local Gorenstein ring. For an ideal I of R let H I(R), i ∈ Z, denote the local cohomology modules of R with respect to I. If H i I(R) = 0 for all i 6= c = height I, then the endomorphism ring of H I (R) is isomorphic to R (cf. [5] and [4]). Here we prove that this is true if and only if H I(R) = 0, i = n, n − 1 provided c ≥ 2 and R/I has an isolated s...

Modules with Dedekind Finite Endomorphism Rings

This article is a survey of modules whose endomorphism rings are Dedekind finite, Hopfian or co-Hopfian. We summarise the properties of such modules and present unified proofs of known results and generalisations to new structure theorems. MSC 2010. 16S50, 16D80.

Endomorphism Rings of Protective Modules

The object of this paper is to study the relationship between certain projective modules and their endomorphism rings. Specifically, the basic problem is to describe the projective modules whose endomorphism rings are (von Neumann) regular, local semiperfect, or left perfect. Call a projective module regular if every cyclic submodule is a direct summand. Thus a ring is a regular module if it is...

Ring With All Finitely Generated Modules Retractable

Direct-sum decompositions of modules with semilocal endomorphism rings

Let R be a ring and C a class of right R-modules closed under finite direct sums. If we suppose that C has a set of representatives, that is, a set V(C) ⊆ C such that every M ∈ C is isomorphic to a unique element [M ] ∈ V(C), then we can view V(C) as a monoid, with the monoid operation [M1] + [M2] = [M1 ⊕M2]. Recent developments in the theory of commutative monoids (e.g., [4], [15]) suggest tha...