Representations of constant socle rank for the Kronecker algebra

Inspired by recent work of Carlson, Friedlander and Pevtsova concerning modules for p-elementary abelian groups {E_{r}} of rank r over a field of characteristic {p>0}, we introduce the notions of modules with constant d-radical rank and modules with constant d-socle rank for the generalized Kronecker algebra {\mathcal{K}_{r}=k\Gamma_{r}} with {r\geq 2} arrows and {1\leq d\leq r-1}. We study subcategories given by modules with the equal d-radical property and the equal d-socle property. Utilizing the simplification method due to Ringel, we prove that these subcategories in {\operatorname{mod}\mathcal{K}_{r}} are of wild type. Then we use a natural functor {\operatorname{\mathfrak{F}}\colon{\operatorname{mod}\mathcal{K}_{r}}\to% \operatorname{mod}kE_{r}} to transfer our results to {\operatorname{mod}kE_{r}}.

The Upper Radical Property and Lower Radical Property of Groups

We take in this paper an arbitrary class [Formula: see text] of groups as a base, and define a radical property 𝒫 for which every group in [Formula: see text] is 𝒫-semisimple. This is called the upper radical property determined by the class [Formula: see text]. At the same time, we define a radical property 𝒫 for which every group in [Formula: see text] is a 𝒫-radical group. This is called the first lower radical property determined by the class [Formula: see text]. Also, we give another construction leading to the second lower radical property which is proved to be identical with the first one.

On Rings Whose Strongly Prime Ideals Are Completely Prime

Kaplansky introduced the concept of the K-rings, concerning the commutativity of rings. In this paper, we concentrate on a property of K-rings, introducing the concept of the strongly NI rings, which is stronger than NI-ness. We first examine the relations among the concepts concerned with K-rings and strongly NI rings, constructing necessary examples in the process. We also show that strong NI-ness is a hereditary radical property.