Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T20:40:46.503Z Has data issue: false hasContentIssue false
  • English
  • Français

Directed graphs and nilpotent rings

Part of: Radicals and radical properties of rings

Published online by Cambridge University Press:  09 April 2009

A. V. Kelarev
A. V. Kelarev
Affiliation:
School of Mathematics, University of Tasmania, G.P.O. Box 252-37, Hobart, Tasmania 7001, Australia e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose that a ring is a sum of its nilpotent subrings. We use directed graphs to give new conditions sufficient for the whole ring to be nilpotent.

MSC classification

Secondary: 16N40: Nil and nilpotent radicals, sets, ideals, rings 16N60: Prime and semiprime rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 3 , December 1998 , pp. 326 - 332
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Bahturin, Yu. A. and Giambruno, A., ‘Identities of sums of commutative subalgebras’ Rend. Circ. Mat. Palermo (2) 43 (1994)(2), 250–258.CrossRefGoogle Scholar
[2]Bahturin, Yu. A. and Kegel, O. H., ‘Lie algebras which are universal sums of abelian subalgebras’, Comm. Algebra 23 (1995), 2975–2990.CrossRefGoogle Scholar
[3]Beidar, K. I. and Mikhalev, A. V., ‘Generalized polynomial identities and rings which are sums of two subrings’, Algebra i Logika 34 (1995)(1), 3–11.Google Scholar
[4]Bokut', L. A., ‘Embeddings in simple associative algebras’, Algebra i Logika 15 (1976) (2), 117–142.Google Scholar
[5]Ferrero, M. and Puczyłowski, E. R., ‘On rings which are sums of two subrings’, Arch. Math. (Basel) 53 (1989), 4–10.CrossRefGoogle Scholar
[6]Fukshansky, A., ‘The sum of two locally nilpotent rings may contain a non-commutative free subring’, Proc. Amer. Math. Soc., to appear.Google Scholar
[7]Herstein, I. N. and Small, L. W., ‘Nil rings satisfying certain chain conditions’, Canad. J. Math. 16 (1964), 771–776.CrossRefGoogle Scholar
[8]Kegel, O. H., ‘Zur Nilpotenz gewisser assoziativer Ringe’, Math. Ann. 149 (1962/1963), 258–260.CrossRefGoogle Scholar
[9]Kegel, O. H., ‘On rings that are sums of two subrings’, J. Algebra 1 (1964), 103–109.CrossRefGoogle Scholar
[10]Kelarev, A. V., ‘A sum of two locally nilpotent rings may be not nil’, Arch. Math. (Basel) 60 (1993), 431–435.CrossRefGoogle Scholar
[11]Kelarev, A. V., ‘A primitive ring which is a sum of two Wedderburn radical subrings’, Proc. Amer. Math. Soc. 125 (1997), 2191–2193.CrossRefGoogle Scholar
[12]Kelarev, A. V., ‘An answer to a question of Kegel on sums of rings’, Canad. Math. Bull. 41 (1998), 79–80.CrossRefGoogle Scholar
[13]Kelarev, A. V. and McConnell, N. R., ‘Two versions of graded rings’, Publ. Math. (Debrecen) 47 (1995) (3–4), 219–227.Google Scholar
[14]Kepczyk, M. and Puczyłowski, E. R., ‘On radicals of rings which are sums of two subrings’. Arch. Math. (Basel) 66 (1996), 8–12.Google Scholar
[15]Kepczyk, M. and Puczyłowski, E. R., ‘Rings which are sums of two subrings’, J. Pure Appl. Algebra. to appear.Google Scholar
[16]Puczyłowski, E. R., ‘Some questions concerning radicals of associative rings’, Theory of Radicals, Szekszárd, 1991, Coll. Math. Soc. János Bolyai 61 (1993), 209–227.Google Scholar
[17]Salwa, A., ‘Rings that are sums of two locally nilpotent subrings’, Comm. Algebra 24 (1996)(12), 3921–3931.CrossRefGoogle Scholar