On Some Artinian QF-3 Rings

Carl Faith in 2003 introduced and investigated an interesting class of rings over which every cyclic right module has Σ-injective injective hull (abbr., right CSI-rings) [5 Faith , C. ( 2003 ). When cyclic modules have Σ-injective hulls . Comm. Algebra 13 : 4161 – 4173 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. Inspired by this we investigate rings over which every cyclic right R-module has a projective Σ-injective injective hull. We show that a ring R satisfies this condition if and only if R is right artinian, the injective hull of R _R is projective and every simple right R-module is embedded in R _R . We also characterize right artinian rings in terms of injective faithful right ideals and right CSI-rings.     

A Note on Relative Flatness and Coherence

Let R be a ring and M a fixed right R-module. A new characterization of M-flatness is given by certain linear equations. For a left R-module F such that the canonical map M? _R F → Hom _R (M?, F) is injective, where M? = Hom _R (M, R), the M-flatness of F is characterized via certain matrix subgroups. An example is given to show that R need not be M-coherent even if every left R-module is M-flat. Moreover, some properties of M-coherent rings are discussed.     

Symmetric Group Decomposition Numbers for Some Three-Part Partitions

The decomposition numbers d _λμ for Specht modules S ^λ of partitions λ with three parts and whose third part is at most p ? 1 are obtained by induction and by using “node removal rules” developed in James and Williams (2000 James , G. D. , Williams , A. L. (2000). Decomposition numbers of symmetric groups by induction. J. Algebra 228:119–142. [CSA] [CROSSREF]  [Google Scholar]).     

On the Twisting and Drinfel'd Double for Multiplier Hopf Algebras

We define the concept of “semiprime” for preradicals and for submodules, and we prove some properties that relate both of them. Related concepts are defined in article by Bican et al. [2 Bican , L. , Jambor , P. , Kepka , T. , Nemec , P. ( 1980 ). Prime and coprime modules . Fundamenta Mathematicae CVII , 33 – 45 . [Google Scholar]] and by Van den Berg and Wisbauer [9 Van den Berg , J. , Wisbauer , R. ( 2001 ). Duprime and dusemiprime modules . Journal of Pure and Applied Algebra 165 : 337 – 356 .[Crossref], [Web of Science ®] , [Google Scholar]]. For any ring, we compare the least semiprime preradical, the Jacobson radical and the join of all nilpotent preradicals, and we characterize V-rings in terms of these three preradicals. We study the least semiprime preradical above any preradical and we prove some of its properties. Using “Amitsur constructions” we define another related operators and prove some of their properties.     

Ganea Term for the Homology of Precrossed Modules

We construct a Ganea term for the homology of precrossed modules, which generalizes the classical Ganea term for the integral homology of groups. We also introduce a central precrossed submodule which relates the Ganea term with capable, unicentral and perfect precrossed modules. Finally, we apply these constructions to the resolution of some open questions in the theory of universal central extensions of precrossed and crossed modules.     

Idealization and Theorems of D. D. Anderson

All rings are commutative with identity and all modules are unital. Anderson proved that a submodule N of an R-module M is multiplication (resp. join principal) if and only if 0₍₊₎ N is a multiplication (resp. join principal) ideal or R(M). The idealization of M. In this article we develop more fully the tool of idealization of a module, particularly in the context of multiplication modules, generalizing Anderson's theorems and discussing the behavior under idealization of some ideals and some submodules associated with a module.     

Amenability and Linear Cellular Automata Over Semisimple Modules of Finite Length

Let M be a semisimple left module of finite length over a ring R and let G be an amenable group. We show that an R-linear cellular automaton τ:M^G → M^G is surjective if and only if it is pre-injective.     

Weakly G K -Perfect and Integral Closure of Ideals

Let R be a commutative Noetherian ring, K a nonzero finitely generated suitable R-module, and I an ideal of R. It is shown that if (R, ) is local, then  is G _K -perfect if and only if K is a canonical module for R. Furthermore, if I is integrally closed and G _K  ? dim _R I < ∞, then K _ is a canonical R _-module for every  ? Ass _R R/I whenever K satisfies Serre's condition (S ₁) or grade _K I > 0. Finally, it is shown that if CM ? dim _R I < ∞, then R _ is Cohen–Macaulay for every  ? Ass _R R/I.