scholarly journals When Intersection Ideals of Graphs of Rings are a Divisor graph

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Local Ring ◽  
Principal Ideal ◽  
Local Rings ◽  
Intersection Graphs ◽  
Principal Ideal Ring ◽  
Empty Intersection ◽  
Ideal Ring

Let R be a commutative principal ideal ring with unity. In this paper, we classify when the intersectiongraphs of ideals of a ring R G(R), is a divisor graph. We prove that the intersection graphs of ideals of a ring RG(R), is a divisor graph if and only if R is a local ring or it is a product of two local rings with each of them hasone chain of ideals. We also prove that G(R), is a divisor graph if it is a product of two local rings one of themhas at most two non-trivial ideals with empty intersection.

Commutative rings whose proper ideals are direct sum of completely cyclic modules

2016 ◽  
Vol 15 (09) ◽  
pp. 1650160 ◽  
Author(s):  
M. Behboodi ◽  
S. Heidari ◽  
S. Roointan-Isfahani
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Principal Ideal ◽  
Simple Formula ◽  
Commutative Rings ◽  
Index Set ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Ideal Ring

By two results of Köthe and Cohen–Kaplansky we obtain that “a commutative ring [Formula: see text] has the property that every [Formula: see text]-module is a direct sum of (completely) cyclic modules if and only if [Formula: see text] is an Artinian principal ideal ring” (an [Formula: see text]-module [Formula: see text] is called completely cyclic if each submodule of [Formula: see text] is cyclic). In this paper, we describe and study commutative rings whose proper ideals are direct sum of completely cyclic modules. It is shown that every proper ideal of a commutative ring [Formula: see text] is a direct sum of completely cyclic [Formula: see text]-modules if and only if [Formula: see text] is a principal ideal ring or [Formula: see text] is a local ring with maximal ideal [Formula: see text] such that there is an index set [Formula: see text] and a set of elements [Formula: see text] such that [Formula: see text] with each [Formula: see text] a simple [Formula: see text]-module and [Formula: see text] a principal ideal ring.

On (α + uβ)-constacyclic codes of length 4ps over 𝔽pm + u𝔽pm∗

2019 ◽  
Vol 18 (02) ◽  
pp. 1950023 ◽  
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta ◽  
Thang M. Vo
Keyword(s):  
Principal Ideal ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Chain Ring ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Ideal Ring ◽  
Unit Formula

For any odd prime [Formula: see text] such that [Formula: see text], the structures of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text] [Formula: see text] are established in term of their generator polynomials. When the unit [Formula: see text] is a square, each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of two constacyclic codes of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that the ambient ring [Formula: see text] is a principal ideal ring. From that, the structure, number of codewords, duals of all such [Formula: see text]-constacyclic codes are obtained. As an application, we identify all self-orthogonal, dual-containing, and the unique self-dual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text].

Special Principal Ideal Rings and Absolute Subretracts

1991 ◽  
Vol 34 (3) ◽  
pp. 364-367 ◽  
Author(s):  
Eric Jespers
Keyword(s):  
Maximal Ideal ◽  
Principal Ideal ◽  
Identity Mapping ◽  
Principal Ideal Ring ◽  
Ideal Ring ◽  
Theoretic Version

AbstractA ring R is said to be an absolute subretract if for any ring S in the variety generated by R and for any ring monomorphism f from R into S, there exists a ring morphism g from S to R such that gf is the identity mapping. This concept, introduced by Gardner and Stewart, is a ring theoretic version of an injective notion in certain varieties investigated by Davey and Kovacs.Also recall that a special principal ideal ring is a local principal ring with nonzero nilpotent maximal ideal. In this paper (finite) special principal ideal rings that are absolute subretracts are studied.

Explicit Formulas Describing: The Binary Operations Calculus In \(E_{a,b}\)

2011 ◽  
Vol 2 (3) ◽  
pp. 16
Author(s):  
Chillali Abdelhakim ◽  
Mohamed Charkani
Keyword(s):  
Elliptic Curve ◽  
Principal Ideal ◽  
Group Homomorphism ◽  
Explicit Formulas ◽  
Principal Ideal Ring ◽  
Binary Operations ◽  
Ideal Ring

In this work we study the elliptic curve over theartinian principal ideal ring \(A=\mathbf{F}_q[\epsilon]\), \((\epsilon^3=0)\). More precisely, we establish a group homomorphism betweens \((\mathbf{F}_q^2,+)\) and theabelian group \(E_{a,b}\) of elliptic curve. For cryptographyapplications, we give meany various explicit formulas describingthe binary operations calculus in \(E_{a,b}\).

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

2019 ◽  
Vol 19 (10) ◽  
pp. 2050185
Author(s):  
Shahabaddin Ebrahimi Atani ◽  
Saboura Dolati Pish Hesari ◽  
Mehdi Khoramdel
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Essential Extension ◽  
Principal Ideal Ring ◽  
Von Neumann ◽  
Simple Modules ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Ideal Ring ◽  
Von Neumann Regular

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.

QUASI-MORPHIC RINGS

2007 ◽  
Vol 06 (05) ◽  
pp. 789-799 ◽  
Author(s):  
V. CAMILLO ◽  
W. K. NICHOLSON
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Ring ◽  
Bezout Ring ◽  
Ideal Ring ◽  
Left And Right ◽  
Regular Rings

A ring R is called left morphic if R/Ra ≅ l (a) for each a ∈ R, equivalently if there exists b ∈ R such that Ra = l (b) and l (a) = Rb. In this paper, we ask only that b and c exist such that Ra = l (b) and l (a) = Rc, and call R left quasi-morphic if this happens for every element a of R. This class of rings contains the regular rings and the left morphic rings, and it is shown that finite intersections of principal left ideals in such a ring are again principal. It is further proved that if R is quasi-morphic (left and right), then R is a Bézout ring and has the ACC on principal left ideals if and only if it is an artinian principal ideal ring.

Rings Whose Modules are ⊕ -Cofinitely Supplemented

2013 ◽  
Vol 0 (0) ◽  
Author(s):  
Ergül Türkmen
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Principal Ideal Ring ◽  
Ideal Ring ◽  
Proper Generalization

Abstract It is known that a commutative ring R is an artinian principal ideal ring if and only if every left R-module is ⊕-supplemented. In this paper, we show that a commutative ring R is a semiperfect principal ideal ring if every left R-module is ⊕-cofinitely supplemented. The converse holds if R is a max ring. Moreover, we study maximally ⊕- supplemented modules as a proper generalization of ⊕-cofinitely supplemented modules. Using these modules, we also prove that a ring R is semiperfect if and only if every projective left R-module with small radical is supplemented.

On ℵα-Noetherian Modules

1970 ◽  
Vol 13 (2) ◽  
pp. 245-247 ◽  
Author(s):  
Aron Simis
Keyword(s):  
Principal Ideal ◽  
Regular Cardinal ◽  
Principal Ideal Ring ◽  
Ideal Ring ◽  
Noetherian Modules

In this note we define two concepts which can be thought of as a generalization of noetherian concepts.The main result is as follows (Corollary A): If R is a ring whose countably generated (left) ideals are (left) principal, then R is a (left) principal ideal ring.This result if obtained, more generally, for any (left) R-module and any regular cardinal ℵα (Corollary 1); a cardinal ℵα is regular whenever W(ℵα) = {ordinals γ | card γ < ℵα} has no cofinal subset of cardinality less than ℵα.

On a Class of Constacyclic Codes of Length 4ps over Fpm[u]〈 ua 〉

2019 ◽  
Vol 26 (02) ◽  
pp. 181-194 ◽  
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta
Keyword(s):  
Principal Ideal ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Chain Ring ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Ideal Ring ◽  
Maximal Ideals

For any odd prime p such that pm ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring [Formula: see text] that have the form Λ = Λ0 + uΛ1 + ⋯ + ua−1 Λa−1, where Λ0, Λ1, …, Λa−1 ∊ 𝔽pm, Λ0 ≠ 0, Λ1 ≠ 0. The class of Λ-constacyclic codes of length 4ps over ℛa is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4ps is expressed as a direct sum of a −λ-constacyclic code and a λ-constacyclic code of length 2ps. In the main case that the unit Λ is not a square, we prove that the polynomial x4 − λ0 can be decomposed as a product of two quadratic irreducible and monic coprime factors, where [Formula: see text]. From this, the ambient ring [Formula: see text] is proven to be a principal ideal ring, whose maximal ideals are ⟨x2 + 2ηx + 2η2⟩ and ⟨x2 − 2ηx + 2η2⟩, where λ0 = −4η4. We also give the unique self-dual Type 1 Λ-constacyclic codes of length 4ps over ℛa. Furthermore, conditions for a Type 1 Λ-constacyclic code to be self-orthogonal and dual-containing are provided.

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