IFP RINGS AND NEAR-IFP RINGS

- Journal title : Journal of the Korean Mathematical Society
- Volume 45, Issue 3, 2008, pp.727-740
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2008.45.3.727

Title & Authors

IFP RINGS AND NEAR-IFP RINGS

Ham, Kyung-Yuen; Jeon, Young-Cheol; Kang, Jin-Woo; Kim, Nam-Kyun; Lee, Won-Jae; Lee, Yang; Ryu, Sung-Ju; Yang, Hae-Hun;

Ham, Kyung-Yuen; Jeon, Young-Cheol; Kang, Jin-Woo; Kim, Nam-Kyun; Lee, Won-Jae; Lee, Yang; Ryu, Sung-Ju; Yang, Hae-Hun;

Abstract

A ring R is called IFP, due to Bell, if ab=0 implies aRb=0 for . Huh et al. showed that the IFP condition need not be preserved by polynomial ring extensions. But it is shown that is a nonzero nilpotent ideal of E whenever R is an IFP ring and is nilpotent, where E is a polynomial ring over R, F is a polynomial ring over E, and are the coefficients of f. we shall use the term near IFP to denote such a ring as having place near at the IFPness. In the present note the structures of IFP rings and near-IFP rings are observed, extending the classes of them. IFP rings are NI (i.e., nilpotent elements form an ideal). It is shown that the near-IFPness and the NIness are distinct each other, and the relations among them and related conditions are examined.

Keywords

IFP ring;near-IFP ring;reduced ring;NI ring;polynomial ring;matrix ring;nilpotent ideal;

Language

English

Cited by

References

1.

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

2.

G. F. Birkenmeier, H. E. Heatherly, and E. K. Lee, Completely prime ideals and associated radicals, Ring theory (Granville, OH, 1992), 102-129, World Sci. Publ., River Edge, NJ, 1993

4.

K. R. Goodearl, von Neumann Regular Rings, Monographs and Studies in Mathematics, 4. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979

5.

K. Y. Ham, C. Huh, Y. C. Hwang, Y. C. Jeon, H. K. Kim, S. M. Lee, Y. Lee, S. R. O, and J. S. Yoon, On weak Armendariz rings, submitted

6.

I. N. Herstein, Topics in Ring Theory, The University of Chicago Press, Chicago-London 1965

7.

C. Y. Hong, H. K. Kim, N. K. Kim, T. K. Kwak, Y. Lee, and K. S. Park, Rings whose nilpotent elements form a Levitzki radical ring, Comm. Algebra 35 (2007), no. 4, 1379-1390

8.

C. Y. Hong and T. K. Kwak, On minimal strongly prime ideals, Comm. Algebra 28 (2000), no. 10, 4867-4878

9.

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

10.

C. Huh, H. K. Kim, D. S. Lee, and Y. Lee, Prime radicals of formal power series rings, Bull. Korean Math. Soc. 38 (2001), no. 4, 623-633

11.

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

12.

S. U. Hwang, Y. C. Jeon, and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), no. 1, 186-199

13.

N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207-223

16.

L. Motais de Narbonne, Anneaux semi-commutatifs et uniseriels; anneaux dont les ideaux principaux sont idempotents, Proceedings of the 106th National Congress of Learned Societies (Perpignan, 1981), 71-73, Bib. Nat., Paris, 1982