We state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. In particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. Also we show that if M is an strong comultiplicati...

we state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. in particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. also we show that if m is an strong comultiplicati...

let $r$ be a commutative ring. in this paper we assert some properties of finitely generated comultiplication modules and fitting ideals of them.

We introduce two constructions of a coassociative comultiplication in the algebra of phrases in a given alphabet. As a preliminary step we give two constructions of a pre-Lie comultiplication in the module generated by words.

Let $R\ $be a commutative ring with $1\neq0$ and $M$ be an $R$-module. Suppose that $S\subseteq R\ $is multiplicatively closed set of $R.\ $Recently Sevim et al. in \cite{SenArTeKo} introduced the notion $S$-prime submodule which is generalization prime used them to characterize certain classes rings/modules such as submodules, simple modules, torsion free modules,\ $S$-Noetherian modules etc. ...

This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n, n) Springer variety. The results build on Khovanov’s work with crossingless matchings and the cohomology of the (n, n) Springer variety. We also give a formula for comultiplication in the Bar-Natan skein module for this specific three-ma...

On path space kQ, there is a trivial kQ-module structure determined by the multiplication of path algebra kQ and a trivial kQ-comodule structure determined by the comultiplication of path coalgebra kQ. In this paper, on path space kQ, a nontrivial kQ-module structure is defined, and it is proved that this nontrivial left kQ-module structure is isomorphic to the dual module structure of trivial ...

Let $R$ be a commutative ring. In this paper we assert some properties of finitely generated comultiplication modules and Fitting ideals of them.

Group algebras are Hopf algebras, and their Hopf structure plays crucial roles in representation theory and cohomology of groups. A Hopf algebra is an algebra A (say over a field k) that has a comultiplication (∆ : A → A ⊗k A) generalizing the diagonal map on group elements, an augmentation (ε : A → k) generalizing the augmentation on a group algebra, and an antipode (S : A → A) generalizing th...

The characterization of H-prime radical is given in many ways. Meantime, the relations between the radical of smash product R#H and the H-radical of Hopf module algebra R are obtained. 0 Introduction and Preliminaries In this paper, let k be a commutative associative ring with unit, H be an algebra with unit and comultiplication △ ( i.e. ∆ is a linear map: H → H ⊗H), R be an algebra over k (R m...