Modules whose partial endomorphisms have a δ-small kernels

Modules Whose Small Submodules Have Krull Dimension

The main aim of this paper is to show that an AB5 module whose small submodules have Krull dimension has a radical having Krull dimension. The proof uses the notion of dual Goldie dimension.

δ-Small Submodules and δ-Supplemented Modules

Contracting Endomorphisms and Gorenstein Modules

A finite module M over a noetherian local ring R is said to be Gorenstein if Ext(k, M) = 0 for all i 6= dimR. A endomorphism φ : R → R of rings is called contracting if φ(m) ⊆ m for some i ≥ 1. Letting R denote the R-module R with action induced by φ, we prove: A finite R-module M is Gorenstein if and only if HomR( R,M) ∼= M and ExtiR( R,M) = 0 for 1 ≤ i ≤ depthR.

Semirings whose additive endomorphisms are multiplicative

A ring or an idempotent semiring is associative provided that additive endomorphisms are multiplicative.

On δ-Quasi Armendariz Modules