ON A GENERALIZATION OF THE MCCOY CONDITION

- Journal title : Journal of the Korean Mathematical Society
- Volume 47, Issue 6, 2010, pp.1269-1282
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2010.47.6.1269

Title & Authors

ON A GENERALIZATION OF THE MCCOY CONDITION

Jeon, Young-Cheol; Kim, Hong-Kee; Kim, Nam-Kyun; Kwak, Tai-Keun; Lee, Yang; Yeo, Dong-Eun;

Jeon, Young-Cheol; Kim, Hong-Kee; Kim, Nam-Kyun; Kwak, Tai-Keun; Lee, Yang; Yeo, Dong-Eun;

Abstract

We in this note consider a new concept, so called -McCoy, which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of -McCoy rings contain upper (lower) triangular matrix rings and many kinds of full matrix rings. We first study the basic structure of -McCoy rings, observing the relations among -McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and ()regular rings. It is proved that the n by n full matrix rings () over reduced rings are not -McCoy, finding -McCoy matrix rings over non-reduced rings. It is shown that the -McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of -McCoy rings are also examined.

Keywords

-McCoy ring;McCoy ring;polynomial ring;matrix ring;classical quotient ring;

Language

English

Cited by

3.

References

1.

D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), no. 6, 2847-2852.

2.

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

3.

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.

V. Camillo and P. P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), no. 3, 599-615.

6.

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

7.

K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, Cambridge, 1989.

8.

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

9.

J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, With the cooperation of L. W. Small. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1987.

10.

N. H. McCoy, Annihilators in polynomial rings, Amer. Math. Monthly 64 (1957), 28-29.