ON QUASI-COMMUTATIVE RINGS

- Journal title : Journal of the Korean Mathematical Society
- Volume 53, Issue 2, 2016, pp.475-488
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2016.53.2.475

Title & Authors

ON QUASI-COMMUTATIVE RINGS

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

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

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;

Language

English

References

2.

D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272.

3.

E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470-473.

4.

H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363-368.

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.

7.

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.

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.

10.

C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative ring, Comm. Algebra 30 (2002), no. 2, 751-761.

11.