The regular digraph of ideals of a commutative ring

AbstractLet$R$be a commutative ring. The regular digraph of ideals of$R$, denoted by$\Gamma (R)$, is a digraph whose vertex set is the set of all nontrivial ideals of$R$and, for every two distinct vertices$I$and$J$, there is an arc from$I$to$J$whenever$I$contains a nonzero divisor on$J$. In this paper, we study the connectedness of$\Gamma (R)$. We also completely characterise the diameter of this graph and determine the number of edges in$\Gamma (R)$, whenever$R$is a finite direct product of fields. Among other things, we prove that$R$has a finite number of ideals if and only if$\mathrm {N}_{\Gamma (R)}(I)$is finite, for all vertices$I$in$\Gamma (R)$, where$\mathrm {N}_{\Gamma (R)}(I)$is the set of all adjacent vertices to$I$in$\Gamma (R)$.

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On quasi-radical of near-ring of polynomials

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2019.56.2.1430 ◽

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.

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N-ideals of commutative rings

Filomat ◽

10.2298/fil1710933t ◽

2017 ◽

Vol 31 (10) ◽

pp. 2933-2941 ◽

Cited By ~ 3

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.

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Derivations of differentially semiprime rings

Asian-European Journal of Mathematics ◽

10.1142/s1793557119500797 ◽

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.

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On the group of unit-valued polynomial functions

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-021-00510-x ◽

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.

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Almost graded multiplication and almost graded comultiplication modules

Demonstratio Mathematica ◽

10.1515/dema-2020-0023 ◽

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.

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Some results on S-primary ideals of a commutative ring

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry ◽

10.1007/s13366-021-00580-5 ◽

2021 ◽

Author(s):

Subramanian Visweswaran

Keyword(s):

Commutative Ring ◽

Primary Ideals

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Artinian bimodule with quasi-Frobenius bimodule of translations

Discrete Mathematics and Applications ◽

10.1515/dma-2019-0010 ◽

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.

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