IJPAM: Volume 113, No. 3 (2017)

Title

$W_1$-ELEMENTS IN A COMMUTATIVE RING WITH UNITY

Authors

W. Nazeer$^1$, W. Khalid$^2$, Shin Min Kang$^3$, W. Ahmad$^4$
$^1$Division of Science and Technology
University of Education
Lahore, 54000, PAKISTAN
$^{2,4}$Department of Mathematics
The University of Lahore
Pakpattan, 57400, PAKISTAN
$^{3}$Department of Mathematics and RINS
Gyeongsang National University
Jinju, 52828, KOREA

Abstract

In this paper, we introduce and study $w_1$-elements in a commutative ring with unity $R$. An element $0 \neq x \in R$ is said to be a $w_1$-element of $R$ if whenever $xd = x$ for $1 \neq d \in R,$ then there exists $0 \neq z \in R$ such that $dz = 0$. We show that if $Ann_R (x) = Ann_R (y)$ then $x$ is a $w_1$-element of $R$ if and only if $y$ is a $w_1$-element of $R,$ that every $w_1$-element of $R$ is also a $w_1$-element of its polynomial ring $R[x]$ and that if $x^2$ is a $w_1$-element of $R$ with $J=Ann_R(x),$ then $x+J$ is a $w_1$-element of its quotient ring $R/J$.

History

Received: December 21, 2016
Revised: February 20, 2017
Published: March 28, 2017
Correction: March 31, 2017, The correspondence author of the article is Professor Shin Min Kang. The original publication is available here.

AMS Classification, Key Words

AMS Subject Classification: 13A15, 13F05
Key Words and Phrases: nilpotent element, annihilator

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Bibliography

1
J.A. Gallian, Contemporary Abstract Algebra, Houghton Mifflin Company (2006).

2
R. Gilmer, Multiplicative Ideal Theory, Pure Appl. Math., 12, Marcel Dekker, Inc, New York (1972).

3
I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago (1974).

How to Cite?

DOI: 10.12732/ijpam.v113i3.10 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 113
Issue: 3
Pages: 483 - 487


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