On quasi-commutative rings

Da Woon Jung; Byung-Ok Kim; Hong Kee Kim; Yang Lee; Sang Bok Nam; Sung Ju Ryu; Hyo Jin Sung; Sang Jo Yun

doi:10.4134/JKMS.2016.53.2.475

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2016; 53(2): 475-488

Printed March 1, 2016

https://doi.org/10.4134/JKMS.2016.53.2.475

On quasi-commutative rings

Da Woon Jung, Byung-Ok Kim, Hong Kee Kim, Yang Lee, Sang Bok Nam, Sung Ju Ryu, Hyo Jin Sung, and Sang Jo Yun

Pusan National University, Korea Science Academy, Gyeongsang National University, Pusan National University, Kyungdong University, Pusan National University, Pusan National University, Pusan National University

Abstract

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Abstract

We study the structure of central elements in relation with polynomial rings and introduce {\it quasi-commutative}as a generalization of commutative rings. The Jacobson radical of the polynomial ring over a quasi-commutative ring is shown to coincide with the set of all nilpotent polynomials; and locally finite quasi-commutative rings are shown to be commutative. We also provide several sorts of examples by showing the relations between quasi-commutative rings and other ring properties which have roles in ring theory. We examine next various sorts of ring extensions of quasi-commutative rings.

Keywords: quasi-commutative ring, polynomial ring, central element, radical

MSC numbers: 16S36, 16N40