[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2016.53.2.475

ON QUASI-COMMUTATIVE RINGS

Jung, Da Woon (Department of Mathematics Pusan National University)
Kim, Byung-Ok (Department of Mathematics Korea Science Academy)
Kim, Hong Kee (Department of Mathematics and RINS Gyeongsang National University)
Lee, Yang (Department of Mathematics Education Pusan National University)
Nam, Sang Bok (Department of Early Child Education Kyungdong University)
Ryu, Sung Ju (Department of Mathematics Pusan National University)
Sung, Hyo Jin (Department of Mathematics Pusan National University)
Yun, Sang Jo (Department of Mathematics Pusan National University)

Publication Information

Journal of the Korean Mathematical Society / v.53, no.2, 2016 , pp. 475-488 More about this Journal

Abstract

We study the structure of central elements in relation with polynomial rings and introduce 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;

Citations & Related Records

Reference

1	S. A. Amitsur, Radicals of polynomial rings, Canad. J. Math. 8 (1956), 355-361. DOI
2	D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272. DOI
3	E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470-473. DOI
4	H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363-368. DOI
5	G. F. Birkenmeier, H. E. Heatherly, and E. K. Lee, Completely prime ideals and associated radicals, Proc. Biennial Ohio State-Denison Conference 1992, 102-129, edited by S. K. Jain and S. T. Rizvi, World Scientific, Singapore-New Jersey-London-Hong Kong, 1993.
6	V. Camillo, C. Y. Hong, N. K. Kim, Y. Lee, and P. P. Nielsen, Nilpotent ideals in polynomial and power series rings, Proc. Amer. Math. Soc. 138 (2010), no. 5, 1607-1619. DOI
7	J. L. Dorroh, Concerning adjunctions to algebras, Bull. Amer. Math. Soc. 38 (1932), no. 2, 85-88. DOI
8	C. Huh, H. K. Kim, and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167 (2002), no. 1, 37-52. DOI
9	C. Huh, H. K. Kim, and Y. Lee, Examples of strongly ${\pi}$ -regular rings, J. Pure Appl. Algebra 189 (2004), no. 1-3, 195-210. DOI
10	C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative ring, Comm. Algebra 30 (2002), no. 2, 751-761. DOI
11	N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488. DOI
12	M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17. DOI