scholarly journals On weakly periodic-like rings and commutativity theorems

2006 ◽  
Vol 37 (4) ◽  
pp. 333-343 ◽  
Author(s):  
Abu-Khuzam Hazar ◽  
Howard E. Bell ◽  
Adil Yaqub
Keyword(s):  
Nilpotent Element ◽  
Commutativity Theorems ◽  
Torsion Freeness ◽  
Positive Integers

A ring $R$ is called periodic if, for every $x$ in $R$, there exist distinct positive integers $m$ and $n$ such that $x^m=x^n$. An element $x$ of $R$ is called potent if $x^k=x$ for some integer $k>1$. A ring $R$ is called weakly periodic if every $x$ in $R$ can be written in the form $x=a+b$ for some nilpotent element $a$ and some potent element $b$ in $R$. A ring $R$ is called weakly periodic-like if every element $x$ in $R$ which is not in the center $C$ of $R$ can be written in the form $x=a+b$, with $a$ nilpotent and $b$ potent. Some structure and commutativity theorems are established for weakly periodic-like rings $R$ satisfying certain torsion-freeness hypotheses along with conditions involving some elements being central.

scholarly journals Commutativity theorems for rings and groups with constraints on commutators

1984 ◽  
Vol 7 (3) ◽  
pp. 513-517 ◽  
Author(s):  
Evagelos Psomopoulos
Keyword(s):  
Associative Ring ◽  
Commutativity Theorems ◽  
Torsion Freeness ◽  
Positive Integers ◽  
Torsion Free

Letn>1,m,t,sbe any positive integers, and letRbe an associative ring with identity. Supposext[xn,y]=[x,ym]ysfor allx,yinR. If, further,Risn-torsion free, thenRis commutativite. Ifn-torsion freeness ofRis replaced by “m,nare relatively prime,” thenRis still commutative. Moreover, example is given to show that the group theoretic analogue of this theorem is not true in general. However, it is true whent=s=0andm=n+1.

Superficial ideals for monomial ideals

2018 ◽  
Vol 17 (06) ◽  
pp. 1850102 ◽  
Author(s):  
Saeed Rajaee ◽  
Mehrdad Nasernejad ◽  
Ibrahim Al-Ayyoub
Keyword(s):  
Noetherian Ring ◽  
Monomial Ideals ◽  
Minimal Set ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Torsion Freeness ◽  
Positive Integers

Let [Formula: see text] and [Formula: see text] be two ideals in a commutative Noetherian ring [Formula: see text]. We say that [Formula: see text] is a superficial ideal for [Formula: see text] if the following conditions are satisfied: (i) [Formula: see text], where [Formula: see text] denotes a minimal set of generators of an ideal [Formula: see text]. (ii) [Formula: see text] for all positive integers [Formula: see text]. In this paper, by using some monomial operators, we first introduce several methods for constructing new ideals which have superficial ideals. In the sequel, we present some examples of monomial ideals which have superficial ideals. Next, we discuss on the relation between superficiality and normality. Finally, we explore the relation between normally torsion-freeness and superficiality.

scholarly journals Two elementary commutativity theorems for generalized boolean rings

1997 ◽  
Vol 20 (2) ◽  
pp. 409-411
Author(s):  
Vishnu Gupta
Keyword(s):  
Positive Integer ◽  
Identity Element ◽  
Negative Integer ◽  
Free Ring ◽  
Boolean Rings ◽  
Commutativity Theorems ◽  
Fixed Positive Integer ◽  
Positive Integers ◽  
Torsion Free

In this paper we prove that ifRis a ring with1as an identity element in whichxm−xn∈Z(R)for allx∈Rand fixed relatively prime positive integersmandn, one of which is even, thenRis commutative. Also we prove that ifRis a2-torsion free ring with1in which(x2k)n+1−(x2k)n∈Z(R)for allx∈Rand fixed positive integernand non-negative integerk, thenRis commutative.

Weakly periodic-like rings and commutativity

2006 ◽  
Vol 43 (3) ◽  
pp. 275-284
Author(s):  
Adil Yaqub
Keyword(s):  
Nilpotent Element ◽  
Positive Integers

A ring R is called periodic if, for every x in R, there exist distinct positive integers m and n such that xm=xn. An element x is called potent if xk=x for some integer k≯1. A ring R is called weakly periodic if every x in R can bewritten in the form x=a+b for some nilpotent element a and some potent element b. A ring R is called weakly periodic-like if every x in R which is not in the center of R can be written in the form x=a+b, where a is nilpotent and b is potent. Our objective is to study the structure of weakly periodic-like rings, with particular emphasis on conditions which yield such rings commutative, or conditions which render the nilpotents N as an ideal of R and R/N as commutative.

scholarly journals Commutativity theorems for rings with constraints on commutators

1991 ◽  
Vol 14 (4) ◽  
pp. 683-688
Author(s):  
Hamza A. S. Abujabal
Keyword(s):  
Polynomial Identity ◽  
Nilpotent Elements ◽  
Zero Divisors ◽  
Commutativity Theorems ◽  
Associative Rings ◽  
Positive Integers

In this paper, we generalize some well-known commutativity theorems for associative rings as follows: Letn>1,m,s, andtbe fixed non-negative integers such thats≠m−1, ort≠n−1, and letRbe a ring with unity1satisfying the polynomial identityys[xn,y]=[x,ym]xtfor ally∈R. Suppose that (i)RhasQ(n)(that isn[x,y]=0implies[x,y]=0); (ii) the set of all nilpotent elements ofRis central fort>0, and (iii) the set of all zero-divisors ofRis also central fort>0. ThenRis commutative. IfQ(n)is replaced by “mandnare relatively prime positive integers,” thenRis commutative if extra constraint is given. Other related commutativity results are also obtained.

Language, procedures, and the non-perceptual origin of natural number concepts

2016 ◽  
Author(s):  
David Barner
Keyword(s):  
General Framework ◽  
Ad Hoc ◽  
Building Blocks ◽  
Number Words ◽  
Linguistic Markers ◽  
Logical Foundations ◽  
Large Numbers ◽  
Perceptual Representations ◽  
Positive Integers ◽  
Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
Yi Zhao
Keyword(s):  
Minimum Size ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

A Note on Differential Identities in Prime and Semiprime Rings

2020 ◽  
pp. 77-83
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeemur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Differential Identities ◽  
Standard Identity ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

scholarly journals The Irregularity and Modular Irregularity Strength of Fan Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 605
Author(s):  
Martin Bača ◽  
Zuzana Kimáková ◽  
Marcela Lascsáková ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Isolated Vertex ◽  
Positive Integers ◽  
Irregularity Strength

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.

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