Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-06T00:05:37.572Z Has data issue: false hasContentIssue false
  • English
  • Français

Solution of the ℐ2-radical problem in structural matrix near-rings

Published online by Cambridge University Press:  14 November 2011

Kirby C. Smith  and
Leon van Wyk
Kirby C. Smith
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368, U.S.A. e-mail: ksmith@math.tamu.edu
Leon van Wyk
Affiliation:
Department of Mathematics, University of Stellenbosch, Private Bag XI, Matieland 7602, South Africa e-mail: lvw@maties.sun.ac.za
Get access Rights & Permissions [Opens in a new window]

Extract

We answer the question, raised in 1989, whether the ℐ2-radical of a structural matrix near-ring can be written as the sum of two nontrivial ideals, one of which is nilpotent, in the affirmative.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 1 , 1998 , pp. 137 - 145
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Abbasi, S. J., Meldrum, J. D. P. and Meyer, J. H.. The ℐ0-radical of matrix near-rings. Arch. Math. Basel 56 (1991), 137–9.CrossRefGoogle Scholar
2Andrunakievic, V. A.. Radicals of associative rings I. Math. Soc. Transl. Ser. 2 52 (1966), 95–128.Google Scholar
3Coelho, S. P.. The automorphism group of a structural matrix algebra. Linear Algebra Appl. 195 (1993), 35–58.CrossRefGoogle Scholar
4Groenewald, N. J. and Wyk, L. van. Polynomial regularities in structural matrix rings. Comm. Algebra 22(1994), 2101–23.CrossRefGoogle Scholar
5Kim, K. H.. Boolean Matrix Theory and Applications (New York: Marcel Dekker, 1982).Google Scholar
6Meldrum, J. D. P.. Near-rings and their Links with Groups, Research Notes in Mathematics 134 (London: Pitman, 1985).Google Scholar
7Meldrum, J. D. P. and Walt, A. P. J. van der. Matrix near-rings. Arch. Math. Basel 47 (1986), 312–19.CrossRefGoogle Scholar
8Meyer, J. H.. Chains of intermediate ideals in matrix near-rings. Arch. Math. Basel 63 (1994), 311–15.CrossRefGoogle Scholar
9Meyer, J. H.. On the near-ring counterpart of the ring isomorphism Mmn(R) ≅ Mn(Mm(R)). Rocky Mountain J. Math., (to appear).Google Scholar
10Pilz, G.. Near-rings, revised edn (Amsterdam: North-Holland, 1983).Google Scholar
11Sands, A. D.. Radicals of structural matrix rings. Quaestiones Math. 13 (1990), 77–81.CrossRefGoogle Scholar
12Walt, A. P. J. van der. Primitivity in matrix near-rings. Quaestiones Math. 9 (1986), 459–69.CrossRefGoogle Scholar
13Walt, A. P. J. van der. On two-sided ideals in matrix near-rings. In Near-rings and Near-fields, ed. Betsch, G., 185–92 (Amsterdam: North-Holland, 1987).Google Scholar
14Walt, A. P. J. van der and Wyk, L. van. The ℐ2-radical in structural matrix near-rings. J. Algebra 123 (1989), 248–61.CrossRefGoogle Scholar
15Wyk, L. van. Subrings of matrix rings (Ph.D. thesis, University of Stellenbosch, 1986).Google Scholar
16Wyk, L. van. Special radicals in structural matrix rings. Comm. Algebra 16 (1988), 421–35.Google Scholar
17Wyk, L. van. The 2-primitive ideals of structural matrix near-rings. Proc. Edinburgh Math. Soc. 34 (1991), 229–39.Google Scholar
18Wyk, L. van. The ℐ2-radical in structural matrix near-rings, II. Proc. Row Soc. Edinburgh Sect. A 122(1992), 53–61.Google Scholar
19Veldsman, S.. On the radicals of structural matrix rings. Monatsh. Math, (to appear).Google Scholar