strongly N-extending Strongly N-extending modules

Relative extending modules and relative (quasi-)continuous modules were introduced and studied by Oshiro as a generalizations of extending modules and (quasi-) continuous respectively. On other hand, Oshiro, Rizvi and Permouth introduced N-extending and N-(quasi-) continuous modules depending where N and M are modules. is closed under submodules, essential extension and isomorphic image. A module M is N-extending if for each submodule A , there is a direct summand B of M such that A is essential in B. Moreover, a module M is strongly extending if every submodule is essential in a stable (equivalently, fully invariant) direct summand of M. In this paper, we introduce and study classes of modules which are proper stronger than that of N-extending modules and N-(quasi-)continuous modules. Many characterizations and properties of these classes are given.

Essential extensions of filters in residuated lattices

We define the essential extension of a filter in the residuated lattice A associated to an ideal of L(A) and investigate its related properties. We prove the residuated lattice A is a Boolean algebra, G(RL)-algebra or MV -algebra if and only if the essential extension of {1} associated to A \ P is a Boolean filter, G-filter or MV -filter (for all P ∈ SpecA), respectively. Also, some properties of lattice of essential extensions are studied.

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.

When is every module with essential socle a direct sum of automorphism-invariant modules?

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if [Formula: see text] is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then [Formula: see text] is right Noetherian. Also, we show a von Neumann regular (semiregular) ring [Formula: see text] with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

Centrally essential rings

A centrally essential ring is a ring which is an essential extension of its center (we consider the ring as a module over its center). We give several examples of noncommutative centrally essential rings and describe some properties of centrally essential rings.

On Injective MV-Modules

In this paper, by considering the notion of MV-module, which is the structure that naturally correspond to lu-modules over lu-rings, we study injective MV-modules and we investigate some conditions for constructing injective MV-modules. Then we define the notions of essential A-homomorphisms and essential extension of A-homomorphisms, where A is a product MV-algebra, and we get some of there properties. Finally, we prove that a maximal essential extension of any A-ideal of an injective MV-module is an injective A-module, too.

W-Closed Submodule and Related Concepts

Let R be a commutative ring with identity, and M be a left untial module. In this paper we introduce and study the concept w-closed submodules, that is stronger form of the concept of closed submodules, where asubmodule K of a module M is called w-closed in M, "if it has no proper weak essential extension in M", that is if there exists a submodule L of M with K is weak essential submodule of L then K=L. Some basic properties, examples of w-closed submodules are investigated, and some relationships between w-closed submodules and other related modules are studied. Furthermore, modules with chain condition on w-closed submodules are studied.

Modules with Chain Conditions on S-Closed Submodules

Let L be a commutative ring with identity and let W be a unitary left L- module. A submodule D of an L- module W is called s- closed submodule denoted by D ≤sc W, if D has no proper s- essential extension in W, that is , whenever D ≤ W such that D ≤se H≤ W, then D = H. In this paper, we study modules which satisfies the ascending chain conditions (ACC) and descending chain conditions (DCC) on this kind of submodules.

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.