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2 results

  • ON $\alpha $-LIKE RADICALS OF RINGS

  • Part of
  • H. FRANCE-JACKSON, T. KHULAN, S. TUMURBAT
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 88 / Issue 2 / October 2013
    Published online by Cambridge University Press:
    12 December 2012, pp. 331-339
    Print publication:
    October 2013
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  • Let $\alpha $ be any radical of associative rings. A radical $\gamma $ is called $\alpha $-like if, for every $\alpha $-semisimple ring $A$, the polynomial ring $A[x] $ is $\gamma $-semisimple. In this paper we describe properties of $\alpha $-like radicals and show how they can be used to solve some open problems in radical theory.

  • ON α-LIKE RADICALS

  • Part of
  • H. FRANCE-JACKSON
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 84 / Issue 1 / August 2011
    Published online by Cambridge University Press:
    16 June 2011, pp. 111-115
    Print publication:
    August 2011
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  • A radical ρ is called prime-like if for every prime ring A, the polynomial ring A[x] is ρ-semisimple. Let α be a radical satisfying the polynomial equation α(A[x])=(α(A))[x] for every ring A. A radical γ is called α-like if for every α-semisimple ring A, the polynomial ring A[x] is γ-semisimple. In this paper, we study properties of α-like radicals. We show that α-likeness is a generalization of prime-likeness and extend some results concerning prime-like radicals. This allows us easily to find distinct special radicals which coincide on simple rings and on polynomial rings, which answers a question put by Ferrero.