The ideal of Lipschitz classical p-compact operators and its injective hull

We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.

On Dense Subsemimodules and Prime Semimodules

In this paper, we study the class of prime semimodules and the related concepts, such as the class of semimodules, the class of Dedekind semidomains, the class of prime semimodules which is invariant subsemimodules of its injective hull, and the compressible semimodules. In order to make the work as complete as possible, we stated, and sometimes proved, some known results related to the above concepts.

Strongly s-dense injective hull and Banaschewski’s theorems for acts

For a class 𝓜 of monomorphisms of a category, mathematicians usually use different types of essentiality. Essentiality is an important notion closely related to injectivity. Banaschewski defines and gives sufficient conditions on a category 𝓐 and a subclass 𝓜 of its monomorphisms under which 𝓜-injectivity well-behaves with respect to the notions such as 𝓜-absolute retract and 𝓜-essentialness. In this paper, 𝓐 is taken to be the category of acts over a semigroup S and 𝓜sd to be the class of strongly s-dense monomorphisms. We study essentiality with respect to strongly s-dense monomorphisms of acts. Depending on a class 𝓜 of morphisms of a category 𝓐, In some literatures, three different types of essentialness are considered. Each has its own benefits in regards with the behavior of 𝓜-injectivity. We will show that these three different definitions of essentiality with respect to the class of strongly s-dense monomorphisms are equivalent. Also, the existence and the explicit description of a strongly s-dense injective hull for any given act which is equivalent to the maximal such essential extension and minimal strongly s-dense injective extension with respect to strongly s-dense monomorphism is investigated. At last we conclude that strongly s-dense injectivity well behaves in the category Act-S.

Injective hulls of many-sorted ordered algebras

This paper is devoted to the study of injectivity for ordered universal algebras. We first characterize injectives in the category $\begin{array}{} \displaystyle {\mathsf{OAL}_{{\it\Sigma}}^{\leqslant}} \end{array}$ of ordered Σ-algebras with lax morphisms as sup-Σ-algebras. Second, we show that every ordered Σ-algebra has an σ⩽-injective hull, and give its concrete form.

Abelian endoregular modules

In this paper, we introduce the notion of abelian endoregular modules as those modules whose endomorphism rings are abelian von Neumann regular. We characterize an abelian endoregular module [Formula: see text] in terms of its [Formula: see text]-generated submodules. We prove that if [Formula: see text] is an abelian endoregular module then so is every [Formula: see text]-generated submodule of [Formula: see text]. Also, the case when the (quasi-)injective hull of a module as well as the case when a direct sum of modules is abelian endoregular are presented. At the end, we study abelian endoregular modules as subdirect products of simple modules.

Prüfer modules over Leavitt path algebras

Let [Formula: see text] denote the Leavitt path algebra associated to the finite graph [Formula: see text] and field [Formula: see text]. For any closed path [Formula: see text] in [Formula: see text], we define and investigate the uniserial, artinian, non-Noetherian left [Formula: see text]-module [Formula: see text]. The unique simple factor of each proper submodule of [Formula: see text] is isomorphic to the Chen simple module [Formula: see text]. In our main result, we classify those closed paths [Formula: see text] for which [Formula: see text] is injective. In this situation, [Formula: see text] is the injective hull of [Formula: see text].

On infinite direct sums of lifting modules

The aim of the present article is to investigate the structure of rings [Formula: see text] satisfying the condition: for any family [Formula: see text] of simple right [Formula: see text]-modules, every essential extension of [Formula: see text] is a direct sum of lifting modules, where [Formula: see text] denotes the injective hull. We show that every essential extension of [Formula: see text] is a direct sum of lifting modules if and only if [Formula: see text] is right Noetherian and [Formula: see text] is hollow. Assume that [Formula: see text] is an injective right [Formula: see text]-module with essential socle. We also prove that if every essential extension of [Formula: see text] is a direct sum of lifting modules, then [Formula: see text] is [Formula: see text]-injective. As a consequence of this observation, we show that [Formula: see text] is a right V-ring and every essential extension of [Formula: see text] is a direct sum of lifting modules for all simple modules [Formula: see text] if and only if [Formula: see text] is a right [Formula: see text]-V-ring.