scholarly journals Modules Whose Nonzero Endomorphisms Have E-small Kernels

2018 ◽  
Vol 10 (3) ◽  
pp. 111
Author(s):  
Abdoul DIALLO ◽  
Papa DIOP ◽  
Mamadou BARRY
Keyword(s):  
Commutative Ring ◽  
Small Submodule

Let $R$ be a commutative ring and  $M$ an unital $R$-module.  A submodule $L$ of $M$ is called essential submodule of $M$, if $L\cap K\neq\lbrace 0\rbrace$ for any nonzero submodule $K$ of $M$. A submodule $N$ of $M$ is called e-small submodule of $M$ if, for any  essential submodule   $L$ of $M$, $N+L= M$  implies $L=M$. An $R$-module $M$ is called e-small quasi-Dedekind module if, for each $f\in End_{R}(M),$ $ f\neq 0$ implies $Kerf$ is e-small in $M$. In this paper we introduce the concept of e-small quasi-Dedekind modules as a generalisation of quasi-Dedekind modules, and give some of their  properties and characterizations.

scholarly journals On S-quasi-Dedekind Modules

2017 ◽  
Vol 9 (5) ◽  
pp. 97 ◽  
Author(s):  
Abdoul Djibril Diallo ◽  
Papa Cheikhou Diop ◽  
Mamadou Barry
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Endomorphism Rings ◽  
Primary Submodule ◽  
Small Submodule

Let $R$ be a commutative ring and $M$ an unital $R$-module. A proper submodule $L$ of $M$ is called primary submodule of $M$, if $rm\in L$, where $r\in R$, $m\in M$, then $m\in L$ or $r^{n}M\subseteq L$ for some positive integer $n$. A submodule $K$ of $M$ is called semi-small submodule of $M$ if, $K+L\neq M$ for each primary submodule $L$ of $M$. An $R$-module $M$ is called S-quasi-Dedekind module if, for each $f\in End_{R}(M),$ $ f\neq 0$ implies $Kerf$ semi-small in $M$. In this paper we introduce the concept of S-quasi-Dedekind modules as a generalisation of small quasi-Dedekind modules, and gives some of their properties, characterizations and exemples. Another hand we study the relationships of S-quasi-Dedekind modules with some classes of modules and their endomorphism rings.

On quasi-radical of near-ring of polynomials

2019 ◽  
Vol 56 (2) ◽  
pp. 252-259
Author(s):  
Ebrahim Hashemi ◽  
Fatemeh Shokuhifar ◽  
Abdollah Alhevaz
Keyword(s):  
Commutative Ring ◽  
Nilpotent Elements ◽  
Ring Of Polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

N-ideals of commutative rings

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2933-2941 ◽  
Author(s):  
Unsal Tekir ◽  
Suat Koc ◽  
Kursat Oral
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

Derivations of differentially semiprime rings

2019 ◽  
Vol 12 (05) ◽  
pp. 1950079
Author(s):  
Ahmad Al Khalaf ◽  
Iman Taha ◽  
Orest D. Artemovych ◽  
Abdullah Aljouiiee
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings ◽  
Semiprime Rings ◽  
Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

scholarly journals On the group of unit-valued polynomial functions

2021 ◽  
Author(s):  
Amr Ali Al-Maktry
Keyword(s):  
Commutative Ring ◽  
Semidirect Product ◽  
Polynomial Functions ◽  
Dual Numbers ◽  
Ring Of Integers ◽  
Group Of Units ◽  
Canonical Representations ◽  
Finite Commutative Ring

AbstractLet R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.

scholarly journals Almost graded multiplication and almost graded comultiplication modules

2020 ◽  
Vol 53 (1) ◽  
pp. 325-331
Author(s):  
Malik Bataineh ◽  
Rashid Abu-Dawwas ◽  
Jenan Shtayat
Keyword(s):  
Commutative Ring ◽  
Graded Modules

AbstractLet G be a group with identity e, R be a G-graded commutative ring with a nonzero unity 1 and M be a G-graded R-module. In this article, we introduce and study the concept of almost graded multiplication modules as a generalization of graded multiplication modules; a graded R-module M is said to be almost graded multiplication if whenever a\in h(R) satisfies {\text{Ann}}_{R}(aM)={\text{Ann}}_{R}(M), then (0{:}_{M}a)=\{0\}. Also, we introduce and study the concept of almost graded comultiplication modules as a generalization of graded comultiplication modules; a graded R-module M is said to be almost graded comultiplication if whenever a\in h(R) satisfies {\text{Ann}}_{R}(aM)={\text{Ann}}_{R}(M), then aM=M. We investigate several properties of these classes of graded modules.

scholarly journals Artinian bimodule with quasi-Frobenius bimodule of translations

2019 ◽  
Vol 29 (2) ◽  
pp. 103-119
Author(s):  
Aleksandr A. Nechaev ◽  
Vadim N. Tsypyschev
Keyword(s):  
Commutative Ring ◽  
Recurrent Sequence ◽  
Artinian Rings ◽  
Linear Recurrent Sequence ◽  
Left And Right ◽  
Additional Assumptions

Abstract The possibility to generalize the notion of a linear recurrent sequence (LRS) over a commutative ring to the case of a LRS over a non-commutative ring is discussed. In this context, an arbitrary bimodule AMB over left- and right-Artinian rings A and B, respectively, is associated with the equivalent bimodule of translations CMZ, where C is the multiplicative ring of the bimodule AMB and Z is its center, and the relation between the quasi-Frobenius conditions for the bimodules AMB and CMZ is studied. It is demonstrated that, in the general case, the fact that AMB is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule CMZ. However, under some additional assumptions it can be shown that if CMZ is a quasi-Frobenius bimodule, then the bimodule AMB is quasi-Frobenius as well.

Some Properties of a Cayley Graph of a Commutative Ring

2013 ◽  
Vol 42 (4) ◽  
pp. 1582-1593 ◽  
Author(s):  
G. Aalipour ◽  
S. Akbari
Keyword(s):  
Commutative Ring ◽  
Cayley Graph
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