Non-Artinian Essential Extensions of Simple Modules

The central simple modules of Artinian Gorenstein algebras

Let A be a standard graded Artinian K-algebra, with char K = 0. We prove the following. 1. A has the Weak Lefschetz Property (resp. Strong Lefschetz Property) if and only if Gr(z)(A) has the Weak Lefschetz Property (resp. Strong Lefschetz Property) for some linear form z of A. 2. If A is Gorenstein, then A has the Strong Lefschetz Property if and only if there exists a linear form z of A such t...

On Minimal Artinian Modules and Minimal Artinian Linear Groups

The paper is devoted to the study of some important types of minimal artinian linear groups. The authors prove that in such classes of groups as hypercentral groups (so also, nilpotent and abelian groups) and FC-groups, minimal artinian linear groups have precisely the same structure as the corresponding irreducible linear groups. 2000 Mathematics Subject Classification. 20E36, 20F28. Let F be ...

Extensions between Simple Modules for Frobenius Kernels

Introduction Let G be a simply connected and connected semisimple group over an algebraically closed field k of characteristic p > 0. T ⊂ G is a maximal torus and R is the root system relative to T . X(T ) is the weight lattice. Let B ⊃ T be a Borel subgroup corresponding to the negative roots R− = R. Denote by Gr the r-th Frobenius kernel of G. The socle and radical structures of the cohomolog...

Extensions of Baer and quasi-Baer modules

On Semi-artinian Weakly Co-semisimple Modules

We show that every semi-artinian module which is contained in a direct sum of finitely presented modules in $si[M]$, is weakly co-semisimple if and only if it is regular in $si[M]$. As a consequence, we observe that every semi-artinian ring is regular in the sense of von Neumann if and only if its simple modules are $FP$-injective.