In this paper we consider module-composed graphs, i.e. graphs which can be defined by a sequence of single vertex insertions v1, . . . , vn, such that the neighbourhood of vertex vi, 2 ≤ i ≤ n, forms a module (a homogenous set) of the graph defined by vertices v1, . . . , vi−1. Module composed graphs are HHDS-free and thus homogeneous orderable, weakly chordal, and perfect. Every bipartite dist...

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with a non-zero identity and $M$ be a graded $R$-module. In this article, we introduce the concept of graded almost semiprime submodules. Also, we investigate some basic properties of graded almost semiprime and graded weakly semiprime submodules and give some characterizations of them.

We give a formula for the relative Deligne tensor product of two indecomposable finite semisimple module categories over pointed braided fusion category an algebraically closed field.

Extending the Wedderburn-Artin theory of (classically) semisimple associative rings to realm topological with right linear topology, we show that abelian category left contramodules over such a ring is split (equivalently, semisimple) if and only discrete modules same semisimple). Our results in this direction complement those Iovanov-Mesyan-Reyes. An extension Bass perfect formulated as list c...

Let G be a connected reductive algebraic group defined over an algebraically closed field of characteristic p > 0. Our first aim in this note is to give concise and uniform proofs for two fundamental and deep results in the context of Serre’s notion of G-complete reducibility, at the cost of less favourable bounds. Here are some special cases of these results: Suppose that the index (H : H◦) is...

The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module M , there exists a module K ∈ σ[M ] such that K ⊕N is weakly injective in σ[M ], for any N ∈ σ[M ]. Similarly, if M is projective and right perfect in σ[M ], then there exists a module K ∈ σ[M ] such that K ⊕ N i...

 Let $R$ be a ring‎, ‎and let $n‎, ‎d$ be non-negative integers‎. ‎A right $R$-module $M$ is called $(n‎, ‎d)$-projective if $Ext^{d+1}_R(M‎, ‎A)=0$ for every $n$-copresented right $R$-module $A$‎. ‎$R$ is called right $n$-cocoherent if every $n$-copresented right $R$-module is $(n+1)$-coprese-nted‎, ‎it is called a right co-$(n,d)$-ring if every right $R$-module is $(n‎, ‎d)$-projective‎. ‎$R$...