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Normalising elements and radicals, I.

Published online by Cambridge University Press:  17 April 2009

E.A. Whelan
Affiliation:
School of Mathematics and Physics, University of East Anglia, Norwich, Norfolk NR4 7TJ, England.
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Abstract

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In this paper we study rings and bimodules with no known one-sided chain conditions, but whose (two-sided) ideals and subbimodules are ‘nicely’ generated. We define bi-noetherian polycentral (BPC) and bi-noetherian polynormal (BPN) rings and bimodules, large classes of (almost always) non-noetherian objects, and put on record the basic facts about them. Any BPC ring is a BPN ring. In the case of rings we reduce their properties to properties of the prime ideals, and study the d.c.c. on (two-sided) ideals. We define both the artinian and bi-artinian radicals of a BPN ring, and use them to show that for BPN rings the intersections of the powers of both the Brown-McCoy and the Jacobson radicals are zero.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Atiyah, M.F. and Macdonald, I.G., Introduction to commutative algebra (Addison-Wesley, Reading, Mass., 1969).Google Scholar
[2]Camina, A.R. and Whelan, E.A., Linear groups and permutations (Pitman, London, 1985).Google Scholar
[3]Chatters, A.W. and Hajarnavis, C.R., Rings with chain conditions (Pitman, London, 1980).Google Scholar
[4]Chatters, A.W., Hajarnavis, C.R. and Norton, N.C., ‘The Artin radical of a Noetherian ring’, J. Austral. Math. Soc. 23 (1977), 379384.CrossRefGoogle Scholar
[5]Chatters, A.W. and Jordan, D.A., ‘Non-commutative unique factorisation rings’, J. London Math. Soc. (2) 33 (1986), 2232.CrossRefGoogle Scholar
[6]Cohen, I.S., ‘Commutative rings with restricetd minimum condition’, Duke Math. J. 17 (1950), 2742.CrossRefGoogle Scholar
[7]Cohen, M. and Montgomery, S., ‘The normal closure of a semiprime ring’, in Ring Theory: proceedings of the 1978 Antwerp conference, van Oystaeyen, F., ed., pp. 4359 (Marcel Dekker, New York, 1979).Google Scholar
[8]Cohn, P.M., Albegra 2 (Wiley, Chichester, 1979).Google Scholar
[9]Divinsky, N.J., Rings and Radicals (George Allen and Unwin, London, 1965).Google Scholar
[10]Heinicke, A.G. and Robson, J.C., ‘Normalising extensions: prime ideals and incomparability’, J. Algebra 72 (1981), 237268.CrossRefGoogle Scholar
[11]Heinicke, A.G. and Robson, J.C., ‘Normalising extensions: prime ideals and incomparability’, University of Leeds, 1983. (preprint)Google Scholar
[12]Jategaonkar, A.V., ‘Localization in noetherian rings’, in L.M.S. Lecture Notes Series 98 (Cambridge Universtiy Press, Cambridge, 1986).Google Scholar
[13]Kaplansky, I., ‘Elementary divisors and modules’, Tans. Amer. Math. Soc. 66 (1949), 464491.CrossRefGoogle Scholar
[14]Lenagan, T.H., ‘Artinian ideals in Noetherian rings’, Proc. Amer. Math. Soc. 51 (1975), 499500.Google Scholar
[15]McConnell, J.C., ‘Localisation in enveloping rings’, J. London Math. Soc. 43 (1968), 421428.CrossRefGoogle Scholar
[16]McLean, K.R., ‘Principal ideal rings and separability’, Proc. London. Math. Soc (3) 45 (1982), 300318.CrossRefGoogle Scholar
[17]Nouazé, Y. and Gabriel, P., ‘Idéaux premiers de l'algèbre enveloppante d'une algèbre de Lie nilpotente’, J. Algebra 6 (1967), 7799.CrossRefGoogle Scholar
[18]Passman, D.S., The algebraic structure of groups and rings (Interscience, New York, 1977).Google Scholar
[19]Robson, J.C., ‘Idealizers and hereditary noetherian prime rings’, J. Algebra 22 (1972), 4581.CrossRefGoogle Scholar
[20]Smith, P.F., ‘On non-commutative regular local rings’, Glasgow Math. J. 17 (1976), 98102.CrossRefGoogle Scholar
[21]Stafford, J.T., ‘On the ideals of a noetherian ring’, University of Leeds, (1984). (preprint)Google Scholar
[22]Whelan, E.A., ‘Quasi-commutative principal ideal rings’, Quart. J. Math. Oxford (2) 37 (1986), 375383.CrossRefGoogle Scholar
[23]Whelan, E.A., ‘Finite subnormalising extensions of rings’, J. Algebra 101 (1986), 418432.CrossRefGoogle Scholar
[24]Whelan, E.A., ‘Symmetry conditions in ring and module theory (Ph.D. Thesis)’, University of East Anglia.Google Scholar
[25]Whelan, E.A., ‘Reduced rank and normalising elements’, J. Algebra 113 (1988), 416429.CrossRefGoogle Scholar
[26]Whelan, E.A., ‘An infinite construction in ring theory’, Glasgow Math. J. 30 (1988), 349357.CrossRefGoogle Scholar
[27]Whelan, E.A., ‘Extensions of rings with normalising elements’. (in preparation)Google Scholar
[28]Whelan, E.A., ‘Normalising elements and radicals II’. (in preparation)Google Scholar
[29]Whelan, E.A., ‘Direct sum decompositions of conformal rings’, Proc. Edinburgh Math. Soc.. (submitted).Google Scholar
[30]Wolfson, K.G., ‘An ideal-theoretic characterisation of the ring of all linear transformations’, Amer. J. Math. 20 (1953), 353386.Google Scholar