Some remarks on one-sided regularity

Abstract

A ring is said to be right (resp., left) regular-duo if every right (resp., left) regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of (one-sided) regular-duo rings, and the relations between one-sided regular-duo rings and related ring theoretic properties.

Keywords

right (left) regular element / right (left) regular-duo ring / upper triangular matrix ring / right (left) Ore domain

References

[1] Antoine R. Nilpotent elements and Armendariz rings. J Algebra, 2008, 319: 3128–3140

[2] Bell H E. Near-rings in which each element is a power of itself. Bull Aust Math Soc, 1970, 2: 363–368

[3] Cohn P M. Reversible rings. Bull Lond Math Soc, 1999, 31: 641–648

[4] Goodearl K R. Von Neumann Regular Rings. London: Pitman, 1979

[5] Huh C, Kim N K, Lee Y. Examples of strongly π-regular rings. J Pure Appl Algebra, 2004, 189: 195–210

[6] Huh C, Lee Y, Smoktunowicz A. Armendariz rings and semicommutative rings. Comm Algebra, 2002, 30: 751–761

[7] Hwang S U, Kim N K, Lee Y. On rings whose right annihilators are bounded. Glasg Math J, 2009, 51: 539–559

[8] Jacobson N. Some remarks on one-sided inverses. Proc Amer Math Soc, 1950, 1: 352–355

[9] Kwak T K, Lee Y, Seo Y. On commutativity of regular products. Bull Korean Math Soc (to appear)

[10] von Neumann J. On regular rings. Proc Natl Acad Sci USA, 1936, 22: 707–713

[11] Nielsen P P. Semi-commutativity and the McCoy condition. J Algebra, 2006, 298: 134–141

[12] Rege M B, Chhawchharia S. Armendariz rings. Proc Japan Acad Ser A Math Sci, 1997, 73: 14–17

RIGHTS & PERMISSIONS

2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature