A Research on the Generalizations of Modules Whose Submodules are Isomorphic to a Direct Summand

A right R-module M is virtually extending (or CIS) if every complement submodule of M is isomorphic to a direct summand of M, and M is called a virtually C2-module if every complement submodule of M which is isomorphic to a direct summand of M is itself a direct summand. The class of virtually extending modules (respectively, virtually C2-modules) is a strict and simultaneous generalization of extending modules (respectively, unifies extending modules and C2-modules): M is a semisimple module if and only if M is virtually semisimple and C2, and M is an extending module if and only if M is virtually extending and virtually C2. Furthermore, every virtually simple right R-module is injective if and only if R is a right V-ring and the class of virtually simple right R-modules coincides with the class of simple right R-modules. Among other results, we show that (1) if all cyclic sub-factors of a cyclic weakly co-Hopfian right R-module M are virtually extending, then M is a finite direct sum of uniform submodules; (2) every distributive virtually extending module over any Noetherian ring is a direct sum of uniform submodules; (3) over a right Noetherian ring, every virtually extending module satisfies the Schröder-Bernstein property; (4) being virtually extending (VC2) is a Morita invariant property; (4) if is a VC2-module where denotes the injective hull, then M is injective. Communicated by Scott Chapman

In this paper, the notation of prime-extending module, is introduced and studied where an Rmodule M is Prime-extending if every nonzero proper submodule of M is essential in prime direct summand .Beside these, the notation of S-prime is given, where an R module is called S-prime if every proper direct summand is prime for this notation we proved: 1.Under the condition of S-prime module, M is Prime-extending if and only if M is extending .2.If M is semisimple module, then M is Prime-extending if and only if M is S-prime.

A module M is called dual-square-free (DSF) if M has no proper submodules A and B with M = A + B and M / A ≅ M / B . The class of DSF-modules is closed under direct summands and homomorphic images, and a module M is distributive iff every submodule of M is DSF. Every DSF-module M is Dedekind finite, and if M satisfies the finite-exchange property, then M satisfies the substitution property and its endomorphism ring has stable range 1. Moreover, a DSF-module M has the finite exchange property iff M is clean, iff M has the full exchange property. Finally, maximal submodules of DSF-modules are shown to be fully-invariant, in particular a ring R is DSF as a right R-module iff R is right quasi-duo (i.e. every maximal right ideal is two-sided). The later result provides a new perspective on the symmetry question of quasi-duo rings.

In this paper, we introduce and study the notions of fuzzy quotient module, fuzzy (simple, semisimple) module and fuzzy maximal submodule. Also, we give many basic properties about these notions.

In this paper we generalize the notion of pure injectivity of modules by introducing what we call a pure Baer injective module. Some properties and some characterization of such modules are established. We also introduce two notions closely related to pure Baer injectivity; namely, the notions of a ∑‐pure Baer injective module and that of SSBI‐ring. A ring R is an SSBI‐ring if and only if every smisimple R‐module is pure Baer injective. To investigate such algebraic structures we had to define what we call p‐essential extension modules, pure relative complement submodules, left pure hereditary rings and some other related notions. The basic properties of these concepts and their interrelationships are explored, and are further related to the notions of pure split modules.

A picture fuzzy set (PFS) is an augmentation of Atanassov’s intuitionistic fuzzy set (IFS). The PFS-based models are useful in the circumstances when we face uncertain and vague information, especially in the case when we need more answers of the form “indeed,” “avoid,” “no,” and “refusal.” It has been considered as an essential tool to deal with unsure data during an investigation. In this manuscript, we explore the idea of a picture fuzzy near-ring (PFNR) and a picture fuzzy ideal (PFI) of a near-ring (NR). We illustrate some basic properties such as union, intersection, homomorphic image, and preimage of PFIs of a NR. Furthermore, there is discussion about the direct product of PFIs of a NR.

Let \(R\) be a ring and \(M\) be an \(R\)-module. \(M\) is called \(\oplus_{ss}\)-supplemented if every submodule of \(M\) has a \(ss\)-supplement that is a direct summand of \(M\). In this paper, the basic properties and characterizations of \(\oplus_{ss}\)-supplemented modules are provided. In particular, it is shown that \((1)\) if a module \(M\) is \(\oplus_{ss}\)-supplemented, then \(Rad(M)\) is semisimple and \(Soc(M)\unlhd M\); \((2)\) every direct sum of \(ss\)-lifting modules is \(\oplus_{ss}\)-supplemented; \((3)\) a commutative ring \(R\) is an artinian serial ring with semisimple radical if and only if every left \(R\)-module is \(\oplus_{ss}\)-supplemented.

In this study, we aimed to introduce two different generalizations of the soft prime int‐ideal and clarify the relationships between the soft prime int‐ideal and the substructures of a ring. First, we explored new algebraic features of the soft prime int‐ideal. We defined and exemplified essential concepts such as soft nilpotent elements, soft idempotent elements, nonexplicit soft int‐ideals, one‐to‐one soft sets, and soft zero divisors of a soft int‐ideal. Then, we investigated the behavior of soft nilpotent and soft idempotent elements in relation to the soft prime int‐ideal. We then defined a soft near‐prime int‐ideal, which is a generalization of the soft prime int‐ideal through soft radicals, and examined its properties. Furthermore, we analyzed the relationships between the soft prime int‐ideal and the soft near‐prime int‐ideal, showing that every soft prime int‐ideal is a soft near‐prime int‐ideal, but the converse does not hold. In addition, we proposed the concept of a soft 1‐absorbing prime int‐ideal as another generalization of the soft prime int‐ideal and studied its basic properties. We proved that the radicals of soft near‐prime int‐ideals and soft 1‐absorbing prime int‐ideals are soft prime int‐ideals. Lastly, we examined soft homomorphic images and preimages of these new structures.

Let B B be a subgroup of an abelian group G G such that G / B G/B is isomorphic to a direct sum of copies of an abelian group A A . For B B to be a direct summand of G G , it is necessary that G G be generated by B B and all homomorphic images of A A in G G . However, if the functor Hom ⁡ ( A , − ) \operatorname {Hom} (A, - ) preserves direct sums of copies of A A , then this condition is sufficient too if and only if M ⊗ E ( A ) A M{ \otimes _{E(A)}}A is nonzero for all nonzero right E ( A ) E(A) -modules M M . Several examples and related results are given.

Weak Amenability of Banach Algebras on Locally Compact Groups

In this chapter, we shall study the central theme of this book, namely, representation of a finite group G over a field K. We establish some of the basic properties using the theory of semi-simple modules over the so called “group algebra” of G over K. (We shall illustrate these results for several important special cases of groups such as S n (Chapter 5), A n (Chapter 6), B n (Chapter 7) and D n (Chapter 8) below.)

Totally Lindelöf and totally ω-narrow paratopological groups