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.

scholarly journals Commutativity results for rings

1988 ◽  
Vol 38 (2) ◽  
pp. 191-195 ◽  
Author(s):  
Hazar Abu-Khuzam
Keyword(s):  
Positive Integer ◽  
Finite Subset ◽  
Associative Ring ◽  
Free Ring ◽  
Commutator Ideal ◽  
Fixed Positive Integer ◽  
Torsion Free

Let R be an associative ring. We prove that if for each finite subset F of R there exists a positive integer n = n(F) such that (xy)n − yn xn is in the centre of R for every x, y in F, then the commutator ideal of R is nil. We also prove that if n is a fixed positive integer and R is an n(n + 1)-torsion-free ring with identity such that (xy)n − ynxn = (yx)n xnyn is in the centre of R for all x, y in R, then R is commutative.

scholarly journals Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Vincenzo De Filippis ◽  
Nadeem UR Rehman ◽  
Abu Zaid Ansari
Keyword(s):  
Lie Ideal ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Free Ring ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers ◽  
Torsion Free

LetRbe a 2-torsion free ring and letLbe a noncentral Lie ideal ofR, and letF:R→RandG:R→Rbe two generalized derivations ofR. We will analyse the structure ofRin the following cases: (a)Ris prime andF(um)=G(un)for allu∈Land fixed positive integersm≠n; (b)Ris prime andF((upvq)m)=G((vrus)n)for allu,v∈Land fixed integersm,n,p,q,r,s≥1; (c)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈[R,R]and fixed integern≥1; and (d)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈Rand fixed integern≥1.

A Combinatorial Theorem

1966 ◽  
Vol 9 (4) ◽  
pp. 515-516
Author(s):  
Paul G. Bassett
Keyword(s):  
Positive Integer ◽  
Combinatorial Theorem ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Positive Integers

Let n be an arbitrary but fixed positive integer. Let Tn be the set of all monotone - increasing n-tuples of positive integers:1Define2In this note we prove that ϕ is a 1–1 mapping from Tn onto {1, 2, 3,…}.

A physical model for teaching multiplication of integers

1968 ◽  
Vol 15 (6) ◽  
pp. 525-528
Author(s):  
Warren H. Hill
Keyword(s):  
Positive Integer ◽  
Physical Model ◽  
Number Line ◽  
Negative Integer ◽  
Physical Models ◽  
Addition And Subtraction ◽  
Positive Integers ◽  
Partial Success

One of the pedagogical pitfalls involved in teaching the multiplication of integers is the scarcity of physical models that can be used to illustrate their product. This problem becomes even more acute if a phyiscal model that makes use of a number line is desired. If the students have previously encountered the operations of addition and subtraction of integers using a number line, then the desirability of developing the operation of multiplication in a similar setting is obvious. Many of the more common physical models that can be used to represent the product of positive and negative integers are unsuitable for interpretation on a number line. Partial success is possible in the cases of the product of two positive integers and the product of a positive and a negative integer. But, how does one illustrate that the product of two negative integers is equal to a positive integer?

The “unknotting number” associated with other local moves than the crossing-change

2018 ◽  
Vol 27 (10) ◽  
pp. 1850051
Author(s):  
Eiji Ogasa
Keyword(s):  
Positive Integer ◽  
Negative Integer ◽  
Positive Integers

The ordinary unknotting number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. Let [Formula: see text] be a positive integer. It is very natural to consider the “unknotting number” associated with other local moves on [Formula: see text]-dimensional knots. In this paper, we prove the following. For the ribbon-move on 2-knots, which is a local move on knots, we have the following: There is a 2-knot which is changed into the unknot by two times of the ribbon-move not by one time. The “unknotting number” associated with the ribbon-move is unbounded. For the pass-move on 1-knots, which is a local move on knots, we have the following: There is a 1-knot such that it is changed into the unknot by two times of the pass-move not by one time and such that the ordinary unknotting number is [Formula: see text]. For any positive integer [Formula: see text], there is a 1-knot whose “unknotting number” associated with the pass-move is [Formula: see text] and whose ordinary unknotting number is [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers. For the [Formula: see text]-move on [Formula: see text]-knots, which is a local move on knots, we have the following: Let [Formula: see text] be a non-negative integer. There is a [Formula: see text]-knot which is changed into the unknot by two times of the [Formula: see text]-move not by one time. The “unknotting number” associated with the [Formula: see text]-move is unbounded. There is a [Formula: see text]-knot which is changed into the unknot by two times of the [Formula: see text]-move not by one. The “unknotting number” associated with the [Formula: see text]-move is unbounded. We prove the following: For any positive integer [Formula: see text] and any positive integer [Formula: see text], there is a [Formula: see text]-knot which is changed into the unknot by [Formula: see text] times of the twist-move not by [Formula: see text] times.

Upper and Lower Bounds for a Function Related to Brown's Lemma

Integers ◽  
2010 ◽  
Vol 10 (6) ◽  
Author(s):  
Hayri Ardal
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Fixed Positive Integer ◽  
Positive Integers

AbstractThe well-known Brown's lemma says that for every finite coloring of the positive integers, there exist a fixed positive integer

scholarly journals On the generalized Ramanujan-Nagell equation $ x^2+(2k-1)^y = k^z $ with $ k\equiv 3 $ (mod 4)

2021 ◽  
Vol 6 (10) ◽  
pp. 10596-10601
Author(s):  
Yahui Yu ◽  
◽  
Jiayuan Hu ◽  
Keyword(s):  
Positive Integer ◽  
Unsolved Problem ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Elementary Methods ◽  
Fixed Positive Integer ◽  
Almost All ◽  
Terai’S Conjecture ◽  
Positive Integers

<abstract><p>Let $ k $ be a fixed positive integer with $ k &gt; 1 $. In 2014, N. Terai <sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup> conjectured that the equation $ x^2+(2k-1)^y = k^z $ has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. This is still an unsolved problem as yet. For any positive integer $ n $, let $ Q(n) $ denote the squarefree part of $ n $. In this paper, using some elementary methods, we prove that if $ k\equiv 3 $ (mod 4) and $ Q(k-1)\ge 2.11 $ log $ k $, then the equation has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. It can thus be seen that Terai's conjecture is true for almost all positive integers $ k $ with $ k\equiv 3 $(mod 4).</p></abstract>

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

scholarly journals SHIFTED AND SHIFTLESS PARTITION IDENTITIES II

2007 ◽  
Vol 03 (01) ◽  
pp. 43-84 ◽  
Author(s):  
FRANK G. GARVAN ◽  
HAMZA YESILYURT
Keyword(s):  
Positive Integer ◽  
Theta Function ◽  
Computer Search ◽  
Modular Functions ◽  
Partition Identities ◽  
Partition Identity ◽  
Function Identity ◽  
Theta Function Identity ◽  
Fixed Positive Integer ◽  
Positive Integers

Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form [Formula: see text] Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M = 32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer search and showed how these could be proved using the theory of modular functions. Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form [Formula: see text] In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four-parameter theta-function identity due to Jacobi. New shifted and shiftless partition identities are proved.

ON n-CENTRALIZING GENERALIZED DERIVATIONS IN SEMIPRIME RINGS WITH APPLICATIONS TO C*-ALGEBRAS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250111 ◽  
Author(s):  
BASUDEB DHARA ◽  
SHAKIR ALI
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
C Algebras ◽  
Semiprime Rings ◽  
Fixed Positive Integer ◽  
Torsion Free

Let R be a ring with center Z(R) and n be a fixed positive integer. A mapping f : R → R is said to be n-centralizing on a subset S of R if f(x)xn – xn f(x) ∈ Z(R) holds for all x ∈ S. The main result of this paper states that every n-centralizing generalized derivation F on a (n + 1)!-torsion free semiprime ring is n-commuting. Further, we prove that if a generalized derivation F : R → R is n-centralizing on a nonzero left ideal λ, then either R contains a nonzero central ideal or λD(Z) ⊆ Z(R) for some derivation D of R. As an application, n-centralizing generalized derivations of C*-algebras are characterized.

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