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On generalised prime essential rings and special and nonspecial radicals

Published online by Cambridge University Press:  17 April 2009

Halina France-Jackson
Halina France-Jackson
Affiliation:
Department of Mathematics and Applied Mathematics, Summerstrand Campus (South), PO Box 77000, Nelson Mandela Metropolitan University, Port Elizabeth 6031, South Africa, e-mail: [email protected]
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For a supernilpotent radical α and a special class σ of rings we call a ring R (α, σ)-essential if R is α-semisimple and for each ideal P of R with R/P ε σ, P ∩ I ≠ 0 whenever I is a nonzero two-sided ideal of R. (α, σ)-essential rings form a generalisation of prime essential rings introduced by L. H. Rowen in his study of semiprime rings and their subdirect decompositions and they have been a subject of investigations of many prominent authors since. We show that many important results concerning prime essential rings are also valid for (α, σ)-essential rings and demonstrate how (α, σ)-essential rings can be used to determine whether a supernilpotent radical is special. We construct infinitely many supernilpotent nonspecial radicals whose semisimple class of prime rings is zero and show that such radicals form a sublattice of the lattice of all supernilpotent radicals. This generalises Yu.M. Ryabukhin's example.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 2 , October 2007 , pp. 263 - 268
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Andrunakievich, V.A. and Ryabukhin, Yu.M., Radicals of algebras and structure theory, (Russian) (Nauka, Moscow, 1979).Google Scholar
[2]France-Jackson, H., ‘On prime essential rings’, Bull. Austral. Math. Soc. 47 (1993), 287–290.CrossRefGoogle Scholar
[3]Gardner, B.J. and Stewart, P.N., ‘Prime essential rings’, Proc. Edinburgh Math. Soc. 34 (1991), 241–250.CrossRefGoogle Scholar
[4]Gardner, B.J. and Wiegandt, R., Radical theory of rings (Marcel Dekker, New York, 2004).Google Scholar
[5]Krachilov, K.K., ‘Coatoms of the lattice of special radicals’, (Russian), Mat. Issled. 49 (1979), 80–86.Google Scholar
[6]Matlis, E., ‘The minimal prime spectrum of a reduced ring’, Illinois J. Math. 27 (1983), 353–391.CrossRefGoogle Scholar
[7]Rowen, L.H., ‘A subdirect decomposition of semiprime rings and its application to maximal quotient rings’, Proc. Amer. Math. Soc. 46 (1974), 176–180.Google Scholar
[8]Ryabukhin, Yu.M., ‘Overnilpotent and special radicals’, Studies in Algebra and Math. Anal., Izdat “Karta Moldovenjaske” Kishinev (1965), 65–72.Google Scholar
[9]Zand, H., ‘A note on prime essential rings’, Bull. Austral. Math. Soc. 49 (1994), 55–57.CrossRefGoogle Scholar