Almost self-injective semimodules

Abstract

A New generalization of injective semimodule has been presented in this working paper. An Ȑ-semimodule Ӎ is called almost self-injective, if Ӎ is almost Ӎ-injective semimodule. Some properties of this notion have been presented. The relationship of this concept to some concepts has also been clarified as End (Ȑ) of indecomposable almost self-injective semimodules, Rad (Ȑ) also some related notions of it have been studied.

Almost Projective Semimodules

“…If Ɲ is Ӎ-injective and every subsemimodule of Ӎ is Ɲ-projective, then every factor of Ɲ is Ӎ-injective semimodule. (2) If every subsemimodule of Ӎ is an Ɲprojective semimodule and every factor of Ɲ is Ӎinjective, then Ɲ is Ӎ-injective.…”

“…In this paper, almost projective modules have been expanded for semimodules taking into account the differences between modules and semimodules, which are mainly derived from their definitions. Almost projective semimodule which is a dual of almost Injective semimodule studied by K. Aljebory and A. Alhossaini 2 . If Ӎ and Ɲ are two semimodule, a semimodule Ӎ is said to be almost Ɲ-projective, if for each epimorphism ϵ: Ɲ ⟶ X where X is any semimodule and every homomorphism 𝛿: Ӎ ⟶ X, either there is a homomorphism 𝜓: Ӎ ⟶ Ɲ such that ϵ 𝜓 = 𝛿, or, there is a homomorphism γ: Y ⟶ Ӎ where Y is a nonzero direct summand of Ɲ such that 𝛿𝛾 = ϵ 𝜆Y where 𝜆Y is the injection map from Y into Ɲ.…”

Some Concepts Related to Almost Injective(Projective)Semi modules

“…Recall the definitions of "almost injective," "almost projective," "almost self-injective," and "almost self-projective" semimodules as in [1], [2], [3], and [4]. Some ideas pertaining to these concepts will be discussed in this research, as well as how they relate to one another.…”

“…The terms "generalizedinjective," "generalized -projective," "essentially -injective," and "length of semimodule" have been defined in this worksheet, and some of their relationships are discussed. If we consider two semimodules Ӑ and Ɓ, then Ӑ is known as almost Ɓ-injective when, for each subsemimodule Ұ of Ɓ and each homomorphism 𝜉: Ұ→ Ӑ , either there exists a homomorphism 𝜁:Ɓ→ Ӑ satisfying 𝜉 = 𝜁𝑖, or there is a homomorphism 𝛾: Ӑ →Ӿ such that 𝛾𝜉 = 𝜋𝑖 where 0 ≠Ӿ ≤ ⨁ Ɓ, and 𝜋 is the projection map [1]. [3] and it is called almost Ɓ-projective, if for each surjective homomorphism α: Ɓ ⟶ Ӿ and every homomorphism 𝛿:Ӑ⟶ Ӿ, either there is 𝜓: Ӑ ⟶ Ɓ where α 𝜓 = 𝛿,or there is γ:Ұ⟶Ӑ where Ұ is nonzero summand of Ɓ such that 𝛿𝛾 = α| Ұ , and Ӑ is called almost-projective if it is almost Ɓ-projective for every finitely generated semimodule Ɓ [2], Ӑ is said to be almost self-projective if it is almost Ӑ-projective [4].…”

Almost Injective Semimodules

“…As regards semimodule, in 1998 Huda Althani gaves an equivalent definition of injective semimodules, which reduces to that in ISSN: 0067-2904 module theory. Also she studied some characterization of injective semimodules [3], later other authors discussed some generalizations of injective semimodules [4 ], [ 5] and [6]. In this work, the concept of injective semimodule has been extended to generalization, almostinjective semimodule.…”