In this paper, we introduce the concept of a quasi-radical semi prime submodule. Throughout work, assume that is commutative ring with identity and left unitary R- module. A proper submodule called (for short Q-rad-semiprime), if for , ,and then . Where intersection all submodules

Let $G$ be an abelian group with identity $e$. $R$ a $G$-graded commutative ring identity, $M$ graded $R$-module and $S\subseteq h(R)$ multiplicatively closed subset of $R$. In this paper, we introduce the concept $S$-prime submodules modules over rings. We investigate some properties class their homogeneous components. $N$ submodule such that $(N:_{R}M)\cap S=\emptyset $. say is \textit{a }$S$...

We provide an upper bound of the dimension of the maximal projective submodule of the Lie module of the symmetric group of n letters in prime characteristic p, where n = pk with p k.

Let l be a prime, and let Γ be a finite subgroup of GLn(Fl) = GL(V ). With these assumptions we say that Condition (C) holds if for every irreducible Γ-submodule W ⊂ ad V there exists an element g ∈ Γ with an eigenvalue α such that tr eg,αW 6= 0. Here, eg,α denotes the projection to the generalised α-eigenspace of g. This condition arises in the definition of adequacy in section 2. Let Γ denote...

Let l be a prime, and let Γ be a finite subgroup of GLn(Fl) = GL(V ). With these assumptions we say that Condition (C) holds if for every irreducible Γ-submodule W ⊂ ad V there exists an element g ∈ Γ with an eigenvalue α such that tr eg,αW 6= 0. Here, eg,α denotes the projection to the generalised α-eigenspace of g. This condition arises in the definition of adequacy in section 2. Let Γ denote...

The goal of this paper is to give a definition generalization fuzzy prime $\Gamma $-ideals in $-rings by introducing 2-absorbing and weakly completely commutative their properties. Furthermore, we diagram which transition between definitions $-ring. Finally, introduce quotient $-ring $R$ induced the $-ideal $2$-absorbing

Let Γ=(V,E) be a graph and ‎W_(‎a)={w_1,…,w_k } be a subset of the vertices of Γ and v be a vertex of it. The k-vector r_2 (v∣ W_a)=(a_Γ (v,w_1),‎…‎ ,a_Γ (v,w_k)) is the adjacency representation of v with respect to W in which a_Γ (v,w_i )=min{2,d_Γ (v,w_i )} and d_Γ (v,w_i ) is the distance between v and w_i in Γ. W_a is called as an adjacency resolving set for Γ if distinct vertices of ...

Mat r x ( F q ) o f a l l r b y r m a t r i c e s o v e r F q , b y C t h e g r o u p G L t i ( F q ) o f t h e u n i t s of M, and by S the special linear subgroup SL , (F q ) o f C . F o r a n a r b i t r a r y field F containing F q , l e t U s t a n d f o r t h e ( c o m m u t a t i v e ) p o l y n o m i a l a l g e b r a F[xl, , x r ] , a n d c o n s i d e r U g r a d e d a s u s u a l : U...

A tag module is a generalization, in any abelian category, of a torsion abelian group. The theory of such modules is developed, it is shown that countably generated tag modules are simply presented, and that Ulm's theorem holds for simply presented tag modules. Zippin's theorem is stated and proved for countably generated tag modules. 1. TAG-modules In the theory of torsion abelian groups, a di...