On Very Large One Sided Ideals of a Ring

1966 ◽  
Vol 9 (2) ◽  
pp. 191-196 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Cardinal Number ◽  
Singular Element

If R is a ring, a right (left) ideal of R is said to be large if it has non-zero intersection with each non-zero right (left) ideal of R [8]. If S is a set, let |S| be the cardinal number of S. We say a right (left) ideal I of a ring R is very large if |R/I| < < No. If a is an element of a ring R such that (a)r = {r ∊ R|ar = 0} is very large then we say a is very singular. The set of all very singular elements of a ring R is a two sided ideal of R. If R is a prime ring, then 0 is the only very singular element of R and a very large right (left) ideal of R is indeed large provided that R is not finite.

AN ENGEL CONDITION WITH AUTOMORPHISMS FOR LEFT IDEALS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350092 ◽  
Author(s):  
CHENG-KAI LIU
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Identity Automorphism ◽  
Semiprime Rings ◽  
Engel Condition ◽  
Positive Integers

Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[…[[σ(xn0), xn1], xn2], …], xnk] = 0 for all x ∈ L, where n0, n1, n2, …, nk are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.

scholarly journals On certain subrings of prime rings with derivations

1993 ◽  
Vol 54 (1) ◽  
pp. 133-141 ◽  
Author(s):  
M. Brešar ◽  
J. Vukman
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
General Result ◽  
Additional Assumption ◽  
Classical Theorem ◽  
Prime Rings

AbstractLet D be a nonzero derivation of a noncommutative prime ring R, and let U be the subring of R generated by all [D(x), x], x ∞ R. A classical theorem of Posner asserts that U is not contained in the center of R. Under the additional assumption that the characteristic of R is not 2, we prove a more general result stating that U contains a nonzero left ideal of R as well as a nonzero right ideal of R.

Centralizing and Commuting Left Generalized Derivations on Prime Rings

2015 ◽  
Vol 11 ◽  
pp. 1-3 ◽  
Author(s):  
C. Jaya Subba Reddy ◽  
S. Mallikarjuna Rao ◽  
V. Vijaya Kumar
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
Prime Rings

Let R be a prime ring and d a derivation on R. If is a left generalized derivation on R such that ƒ is centralizing on a left ideal U of R, then R is commutative.

scholarly journals A Note on a Self Injective Ring

1965 ◽  
Vol 8 (1) ◽  
pp. 29-32 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Minimum Condition ◽  
Maximum Condition ◽  
Simple Ring ◽  
The Right ◽  
Primitive Ring

A ring R with unity is called right (left) self injective if the right (left) R-module R is injective [7]. The purpose of this note is to prove the following: Let R be a prime ring with a maximal annihilator right (left) ideal. If R is right (left) self injective then R is a primitive ring with a minimal one-sided ideal. If R satisfies the maximum condition on annihilator right (left) ideals and R is right (left) self injective then R is a simple ring with the minimum condition on one-sided ideals.

Generalized Derivations Having the Same Power Values with Left Multiplications

2013 ◽  
Vol 20 (03) ◽  
pp. 369-382 ◽  
Author(s):  
Xiaowei Xu ◽  
Jing Ma ◽  
Fengwen Niu
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Prime Ring ◽  
Generalized Derivation ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Ring Of Quotients ◽  
Fixed Positive Integer

Let R be a prime ring with extended centroid C, maximal right ring of quotients U, a nonzero ideal I and a generalized derivation δ. Suppose δ(x)n =(ax)n for all x ∈ I, where a ∈ U and n is a fixed positive integer. Then δ(x)=λax for some λ ∈ C. We also prove two generalized versions by replacing I with a nonzero left ideal [Formula: see text] and a noncommutative Lie ideal L, respectively.

scholarly journals A Note on a Prime Ring with a Maximal Annihilator Right Ideal

1965 ◽  
Vol 8 (1) ◽  
pp. 109-110 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Present Note ◽  
Zero Element

A ring R is called a prime ring [1] if and only if a·R·b = 0 implies that a = 0 or b = 0 for all a, b ϵ R. Hence if R is a prime ring and a is a non-zero element of R, a·R ≠ 0 and R·a ≠ 0. In the present note we prove that a prime ring with a maximal annihilator right ideal has no non-zero nil right or left ideal.

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.

Reconstruction of the Ancient Numeral System in Basque Language

Philology ◽  
2019 ◽  
Vol 4 (2018) ◽  
pp. 157-172
Author(s):  
FERNANDO GOMEZ-ACEDO ◽  
ENEKO GOMEZ-ACEDO
Keyword(s):  
Cardinal Number ◽  
Counting System ◽  
Cardinal Numbers ◽  
Original Meaning ◽  
Diachronic Development ◽  
Insight Into

Abstract In this work a new insight into the reconstruction of the original forms of the first Basque cardinal numbers is presented and the identified original meaning of the names given to the numbers is shown. The method used is the internal reconstruction, using for the etymologies words that existed and still exist in Basque and other words reconstructed from the proto-Basque. As a result of this work it has been discovered that initially the numbers received their name according to a specific and logic procedure. According to this ancient method of designation, each cardinal number received its name based on the hand sign used to represent it, thus describing the position adopted by the fingers of the hand to represent each number. Finally, the different stages of numerical formation are shown, which demonstrate a long and diachronic development of the whole counting system.

m-potent commutators involving skew derivations and multilinear polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Ashraf ◽  
Sajad Ahmad Pary ◽  
Mohd Arif Raza
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Prime Rings ◽  
Skew Derivation ◽  
Martindale Quotient Ring ◽  
Multilinear Polynomials ◽  
The Right ◽  
The Relationship

AbstractLet {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1<m\in\mathbb{Z}^{+}}, {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over {\mathscr{C}} and δ is a skew derivation of {\mathscr{R}}.

IMPRECISE CLASSIFICATION WITH CREDAL DECISION TREES

2012 ◽  
Vol 20 (05) ◽  
pp. 763-787 ◽  
Author(s):  
JOAQUÍN ABELLÁN ◽  
ANDRÉS R. MASEGOSA
Keyword(s):  
Decision Trees ◽  
Naive Bayes ◽  
Cardinal Number ◽  
Naïve Bayes ◽  
Classification Method ◽  
Cost Matrix ◽  
Imprecise Dirichlet Model ◽  
Dirichlet Model

In this paper, we present the following contributions: (i) an adaptation of a precise classifier to work on imprecise classification for cost-sensitive problems; (ii) a new measure to check the performance of an imprecise classifier. The imprecise classifier is based on a method to build simple decision trees that we have modified for imprecise classification. It uses the Imprecise Dirichlet Model (IDM) to represent information, with the upper entropy as a tool for splitting. Our new measure to compare imprecise classifiers takes errors into account. Thus far, this has not been considered by other measures for classifiers of this type. This measure penalizes wrong predictions using a cost matrix of the errors, given by an expert; and it quantifies the success of an imprecise classifier based on the cardinal number of the set of non-dominated states returned. To compare the performance of our imprecise classification method and the new measure, we have used a second imprecise classifier known as Naive Credal Classifier (NCC) which is a variation of the classic Naive Bayes using the IDM; and a known measure for imprecise classification.

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