In this article we study injective representations of infinite quivers. We classify the indecomposable injective representations of trees and then describe Gorenstein injective and projective representations of barren trees.

We give some new properties of almost injective modules and their endomorphism rings, and also provide conditions as to when a direct sum of almost injective (or CS) modules is again almost injective (or CS) in some special cases..

Lemma 1. There are ω1 disjoint stationary subsets of ω1. Proof. Let Cub(ω1) = {X ⊆ ω1 : ∃C ⊆ X(C is club on ω1)}. Then Cub(ω1) is countably complete filter. Its dual ideal Cub (ω1) = {X ⊆ ω1 : ∃X ′ ∈ Cub(ω1)(X = ω1\X ′)} is countably complete and contains all singletons and so all countable subsets of ω1. Recall also that X ⊆ ω1 is stationary if and only if X / ∈ Cub(ω1). For every ρ < ω1 let f...

The Jacobian Conjecture would follow if it were known that real polynomial maps with a unipotent Jacobian matrix are injective. The conjecture that this is true even for C maps is explored here. Some results known in the polynomial case are extended to the C context, and some special cases are resolved.

In any concrete category C the natural problem of whether epimorphisms are surjective maps arise, as well as the dual problem of whether the monomorphisms are injective maps. This type of problems have already been studied before in several well known categories: for example in [6] it is shown that the property of epimorphisms of being surjective holds in the categories of von Neumann algebras,...

and Applied Analysis 3 2. α-Times Regularized Resolvent Family Throughout this paper, X is a complex Banach space, and we denote by B X the algebra of all bounded linear operators on X. Let A be a closed densely defined operator on X, let D A and R A be its domain and range, respectively, and let α ∈ 0, 2 , C ∈ B X be injective. Define ρC A : {λ ∈ C : λ − A is injective and R C ⊂ R λ − A }. Let...

In act theory, Pseudo injective acts and their generalizations are essential. As a result, the purpose of this work is to give generalization pseudo quasi principally specifically small acts. Besides, concept was presented, which can be employed later. If each S-monomorphism from M-cyclic sub-act M\textsubscript{S} N\textsubscript{S} extended S-homomorphism N\textsubscript{S}, An S-act called p...

The notion of simple-direct-injective modules which are a generalization injective unifies $C2$ and $C3$-modules. In the present paper, we introduce semisimple-direct-injective module gives unified viewpoint $C2$, $C3$, SSP properties modules. It is proved that ring $R$ Artinian serial with Jacobson radical square zero if only every right $R$-module has and, for any family simple $R$-modules $\...

In this paper, we study the relation between m-strongly Gorenstein projective (resp. injective) modules and n-strongly Gorenstein projective (resp. injective) modules whenever m 6= n, and the homological behavior of n-strongly Gorenstein projective (resp. injective) modules. We introduce the notion of n-strongly Gorenstein flat modules. Then we study the homological behavior of n-strongly Goren...