An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers

Let R=(R,+,·) be a commutative ring of characteristic m>0 (m may be equal to +∞) with unity e and zero 0. Given a positive integer n<m and the so-called n-symmetric set A=a1,a2,…,a2l-1,a2l such that al+i=ne-ai for each i=1,…,l, define the rth power sum Sr(A) as Sr(A)=∑i=12lair, for r=0,1,2,…. We prove that for each positive integer k there holds ∑i=02k-1(-1)i2k-1i22k-1-iniS2k-1-i(A)=0. As an application, we obtain two new Pascal-like identities for the sums of powers of the first n-1 positive integers.

On 1-absorbing primary ideals of commutative rings

10.1142/s021949882050111x ◽

2019 ◽

Vol 19 (06) ◽

pp. 2050111 ◽

Cited By ~ 2

Author(s):

Ayman Badawi ◽

Ece Yetkin Celikel

Keyword(s):

Positive Integer ◽

Prime Ideal ◽

Commutative Ring ◽

Commutative Rings ◽

Proper Ideal ◽

Primary Ideal ◽

Ideal Formula ◽

Noetherian Domain ◽

Primary Ideals ◽

Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-absorbing primary ideal of [Formula: see text] if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some properties of 1-absorbing primary ideals are investigated. For example, we show that if [Formula: see text] admits a 1-absorbing primary ideal that is not a primary ideal, then [Formula: see text] is a quasilocal ring. We give an example of a 1-absorbing primary ideal of [Formula: see text] that is not a primary ideal of [Formula: see text]. We show that if [Formula: see text] is a Noetherian domain, then [Formula: see text] is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of [Formula: see text] is of the form [Formula: see text] for some nonzero prime ideal [Formula: see text] of [Formula: see text] and a positive integer [Formula: see text]. We show that a proper ideal [Formula: see text] of [Formula: see text] is a 1-absorbing primary ideal of [Formula: see text] if and only if whenever [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text], then [Formula: see text] or [Formula: see text]

DEFINING POWER SUMS OF n AND φ(n) INTEGERS

International Journal of Number Theory ◽

10.1142/s179304210900189x ◽

2009 ◽

Vol 05 (01) ◽

pp. 41-53 ◽

Cited By ~ 7

Author(s):

JITENDER SINGH

Keyword(s):

Positive Integer ◽

Closed Form ◽

Power Sums ◽

Power Sum ◽

Analytical Results ◽

General Power ◽

Integral Relations ◽

Let n be a positive integer and φ(n) denotes the Euler phi function. It is well known that the power sum of n can be evaluated in closed form in terms of n. Also, the sum of all those φ(n) positive integers that are coprime to n and not exceeding n, is expressible in terms of n and φ(n). Although such results already exist in literature, but here we have presented some new analytical results in these connections. Some functional and integral relations are derived for the general power sums.

On the symmetric digraphs from the kth power mapping on finite commutative rings

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830914500645 ◽

2015 ◽

Vol 07 (01) ◽

pp. 1450064 ◽

Cited By ~ 1

Author(s):

Guixin Deng ◽

Lawrence Somer

Keyword(s):

Positive Integer ◽

Commutative Ring ◽

Sufficient Conditions ◽

Commutative Rings ◽

Connected Components ◽

Directed Edge ◽

Finite Commutative Ring ◽

Power Mapping ◽

Factor G ◽

Finite Commutative Rings

For a finite commutative ring R and a positive integer k, let G(R, k) denote the digraph whose set of vertices is R and for which there is a directed edge from a to ak. The digraph G(R, k) is called symmetric of order M if its set of connected components can be partitioned into subsets of size M with each subset containing M isomorphic components. We primarily aim to factor G(R, k) into the product of its subdigraphs. If the characteristic of R is a prime p, we obtain several sufficient conditions for G(R, k) to be symmetric of order M.

A note on reductions of ideals relative to an Artinian module

Glasgow Mathematical Journal ◽

10.1017/s0017089500009770 ◽

1993 ◽

Vol 35 (2) ◽

pp. 219-224 ◽

Cited By ~ 2

Author(s):

A.-J. Taherizadeh

Keyword(s):

Positive Integer ◽

Commutative Ring ◽

Integral Closure ◽

Artinian Modules ◽

Artinian Module ◽

Image Position ◽

The concept of reduction and integral closure of ideals relative to Artinian modules were introduced in [7]; and we summarize some of the main aspects now.Let A be a commutative ring (with non-zero identity) and let a, b be ideals of A. Suppose that M is an Artinian module over A. We say that a is a reduction of b relative to M if a ⊆ b and there is a positive integer s such that)O:Mabs)=(O:Mbs+l).An element x of A is said to be integrally dependent on a relative to M if there exists n y ℕ(where ℕ denotes the set of positive integers) such thatIt is shown that this is the case if and only if a is a reduction of a+Ax relative to M; moreoverᾱ={x ɛ A: xis integrally dependent on a relative to M}is an ideal of A called the integral closure of a relative to M and is the unique maximal member of℘ = {b: b is an ideal of A which has a as a reduction relative to M}.

Vertex and region colorings of planar idempotent divisor graphs of commutative rings.

Iraqi Journal for Computer Science and Mathematics ◽

10.52866/ijcsm.2022.01.01.008 ◽

2022 ◽

pp. 71-82

Author(s):

Mohammed Authman ◽

Husam Q. Mohammad ◽

Nazar H. Shuker

Keyword(s):

Graph Theory ◽

Positive Integer ◽

Prime Number ◽

Commutative Ring ◽

Chromatic Number ◽

Clique Number ◽

Commutative Rings ◽

Ring Theory ◽

Idempotent Element

The idempotent divisor graph of a commutative ring R is a graph with vertices set in R* = R-{0}, and any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e2 = e ϵ R, and is denoted by Л(R). The purpose of this work is using some properties of ring theory and graph theory to find the clique number, the chromatic number and the region chromatic number for every planar idempotent divisor graphs of commutative rings. Also we show the clique number is equal to the chromatic number for any planar idempotent divisor graph. Among other results we prove that: Let Fq, Fpa are fieldes of orders q and pa respectively, where q=2 or 3, p is a prime number and a Is a positive integer. If a ring R @ Fq x Fpa . Then (Л(R))= (Л(R)) = *( Л(R)) = 3.

On n-absorbing and strongly n-absorbing ideals of amalgamation

10.1142/s0219498820501996 ◽

2019 ◽

Vol 19 (10) ◽

pp. 2050199

Author(s):

Mohammed Issoual ◽

Najib Mahdou ◽

Moutu Abdou Salam Moutui

Keyword(s):

Positive Integer ◽

Prime Ideal ◽

Commutative Ring ◽

Commutative Rings ◽

Ring Homomorphism ◽

Proper Ideal ◽

Prime Ideals ◽

Ideal Formula

Let [Formula: see text] be a commutative ring with [Formula: see text]. Let [Formula: see text] be a positive integer. A proper ideal [Formula: see text] of [Formula: see text] is called an n-absorbing ideal (respectively, a strongly n-absorbing ideal) of [Formula: see text] as in [D. F. Anderson and A. Badawi, On [Formula: see text]-absorbing ideals of commutative rings, Comm. Algebra 39 (2011) 1646–1672] if [Formula: see text] and [Formula: see text], then there are [Formula: see text] of the [Formula: see text]’s whose product is in [Formula: see text] (respectively, if whenever [Formula: see text] for ideals [Formula: see text] of [Formula: see text], then the product of some [Formula: see text] of the [Formula: see text]s is contained in [Formula: see text]). The concept of [Formula: see text]-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of [Formula: see text] is a 1-absorbing ideal of [Formula: see text]). Let [Formula: see text] be a ring homomorphism and let [Formula: see text] be an ideal of [Formula: see text] This paper investigates the [Formula: see text]-absorbing and strongly [Formula: see text]-absorbing ideals in the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect [Formula: see text] denoted by [Formula: see text] The obtained results generate new original classes of [Formula: see text]-absorbing and strongly [Formula: see text]-absorbing ideals.

On (m,n)-closed ideals of commutative rings

10.1142/s021949881750013x ◽

2017 ◽

Vol 16 (01) ◽

pp. 1750013 ◽

Cited By ~ 5

Author(s):

David F. Anderson ◽

Ayman Badawi

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Closed Ideal ◽

Proper Ideal ◽

Text Recall ◽

Let [Formula: see text] be a commutative ring with [Formula: see text], and let [Formula: see text] be a proper ideal of [Formula: see text]. Recall that [Formula: see text] is an [Formula: see text]-absorbing ideal if whenever [Formula: see text] for [Formula: see text], then there are [Formula: see text] of the [Formula: see text]’s whose product is in [Formula: see text]. We define [Formula: see text] to be a semi-[Formula: see text]-absorbing ideal if [Formula: see text] for [Formula: see text] implies [Formula: see text]. More generally, for positive integers [Formula: see text] and [Formula: see text], we define [Formula: see text] to be an [Formula: see text]-closed ideal if [Formula: see text] for [Formula: see text] implies [Formula: see text]. A number of examples and results on [Formula: see text]-closed ideals are discussed in this paper.

On locating numbers and codes of zero divisor graphs associated with commutative rings

10.1142/s0219498816500146 ◽

2015 ◽

Vol 15 (01) ◽

pp. 1650014 ◽

Cited By ~ 7

Author(s):

Rameez Raja ◽

S. Pirzada ◽

Shane Redmond

Keyword(s):

Commutative Ring ◽

Connected Graph ◽

Zero Divisor ◽

Domination Number ◽

Clique Number ◽

Commutative Rings ◽

Equivalence Relations ◽

Finite Product ◽

Positive Integers ◽

The Relationship

Let R be a commutative ring with identity and let G(V, E) be a graph. The locating number of the graph G(V, E) denoted by loc (G) is the cardinality of the minimal locating set W ⊆ V(G). To get the loc (G), we assign locating codes to the vertices V(G)∖W of G in such a way that every two vertices get different codes. In this paper, we consider the ratio of loc (G) to |V(G)| and show that there is a finite connected graph G with loc (G)/|V(G)| = m/n, where m < n are positive integers. We examine two equivalence relations on the vertices of Γ(R) and the relationship between locating sets and the cut vertices of Γ(R). Further, we obtain bounds for the locating number in zero-divisor graphs of a commutative ring and discuss the relation between locating number, domination number, clique number and chromatic number of Γ(R). We also investigate the locating number in Γ(R) when R is a finite product of rings.

Certain near-rings are rings, II

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171286000327 ◽

1986 ◽

Vol 9 (2) ◽

pp. 267-272 ◽

Cited By ~ 2

Author(s):

Howard E. Bell

Keyword(s):

Positive Integer ◽

Commutative Ring ◽

We investigate distributively-generated near-ringsRwhich satisfy one of the following conditions: (i) for eachx,y∈R, there exist positive integersm,nfor whichxy=ymxn; (ii) for eachx,y∈R, there exists a positive integernsuch thatxy=(yx)n. Under appropriate additional hypotheses, we prove thatRmust be a commutative ring.

On n-irreducible ideals of commutative rings

10.1142/s0219498820501200 ◽

2019 ◽

Vol 19 (06) ◽

pp. 2050120

Author(s):

Nabil Zeidi

Keyword(s):

Positive Integer ◽

Commutative Ring ◽

Commutative Rings ◽

Proper Ideal ◽

Ideal Formula ◽

Strongly Irreducible

Let [Formula: see text] be a commutative ring with [Formula: see text] and [Formula: see text] a positive integer. The main purpose of this paper is to study the concepts of [Formula: see text]-irreducible and strongly [Formula: see text]-irreducible ideals which are generalizations of irreducible and strongly irreducible ideals, respectively. A proper ideal [Formula: see text] of [Formula: see text] is called [Formula: see text]-irreducible (respectively, strongly [Formula: see text]-irreducible) if for each ideals [Formula: see text] of [Formula: see text], [Formula: see text] (respectively, [Formula: see text]) implies that there are [Formula: see text] of the [Formula: see text]’s whose intersection is [Formula: see text] (respectively, whose intersection is in [Formula: see text]).