NIL-REFLEXIVE RINGS

Year 2016, Volume: 65 Issue: 1, 19 - 34, 01.02.2016

Handan Kose Burcu Ungor Abdullah Harmancı

https://doi.org/10.1501/Commua1_0000000741

Cited By: 1

Abstract

In this paper, we deal with a new approach to reflexive property
for rings by using nilpotent elements, in this direction we introduce nil-reflexive
rings. It is shown that the notion of nil-reflexive is a generalization of that
of nil-semicommutativity. Examples are given to show that nil-reflexive rings
need not be reflexive and vice versa, and nil-reflexive rings but not semicommutative are presented. We also proved that every ring with identity is weakly
reflexive defined by Zhao, Zhu and Gu. Moreover, we investigate basic properties of nil-reflexive rings and provide some source of examples for this class
of rings. We consider some extensions of nil-reflexive rings, such as trivial
extensions, polynomial extensions and Nagata extensions.

Keywords

nil-semicommutative ring

References

• F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1992.
• R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra, 319(8)(2008), 3128-3140.
• W. Chen, On nil-semicommutative rings, Thai J. Math., 9(1)(2011), 39-47.
• P. M. Cohn, Reversible rings, Bull. London Math. Soc., 31(6)(1999), 641-648.
• N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra, 185(2003), 223.
• T. K. Kwak and Y. Lee, Re*exive property of rings, Comm. Algebra, 40(2012), 1576-1594.
• T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 2001.
• J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull., 14(1971), 359-368.
• G. Mason, Re*exive ideals, Comm. Algebra, 9(17)(1981), 1709-1724.
• R. Mohammadi, A. Moussavi and M. Zahiri, On nil-semicommutative rings, Int. Electron. J. Algebra, 11(2012), 20-37.
• M. Nagata, Local Rings, Interscience, New York, 1962.
• M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci., (1)(1997), 14-17.
• L. Zhao, X. Zhu and Q. Gu, Re*exive rings and their extensions, Math. Slovaca, 63(3)(2013), 430.